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b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(58)785 995 y(3.5)94 b(Pro)s(of)30 b(of)g(Bullsey)m(e)g(Sp)s(ecial)f(Case)60 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)g(.)f(.)h(.)108 b(61)785 1108 y(3.6)94 b(Pro)s(of)30 b(of)g(Pure)g(Sp)s(ecial)e(Case)58 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(63)785 1220 y(3.7)94 b(Pro)s(of)30 b(of)g(Symmetric)f(Case)60 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(68)785 1333 y(3.8)94 b(Pro)s(of)30 b(of)g(General)g(Case)36 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f (.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.) h(.)108 b(72)785 1446 y(3.9)94 b(Examples)29 b(of)i Fq(QAP)p Fw('s)e(and)h(their)f(Cen)m(tral)h(Pro)5 b(jections)55 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(74)504 1672 y Fs(IV.)42 b(Concluding)35 b(Remarks)47 b Fw(.)f(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.) f(.)h(.)g(.)g(.)f(.)h(.)108 b(79)785 1898 y(4.1)94 b(Related)30 b(Optimization)f(Problems)77 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g (.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(79)1085 2011 y(4.1.1)d(The)30 b(Linear)g(Assignmen)m(t)g(Problem)83 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(79)1085 2124 y(4.1.2)d(Higher-dimensional)28 b(Assignmen)m(t)i (Problems)75 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(80)1085 2237 y(4.1.3)d(The)30 b(BiQuadratic)g(Assignmen)m(t)f (Problem)61 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(80)1085 2350 y(4.1.4)d(The)30 b(W)-8 b(eigh)m(ted)32 b(Hyp)s(ergraph)d(Matc)m(hing)i(Problem)45 b(.)g(.)h(.)g(.)g(.)f(.)h(.) 108 b(81)785 2462 y(4.2)94 b(Notes)31 b(on)f(Heuristics)69 b(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.) g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(81)1085 2575 y(4.2.1)d(Pro)s(ofs)30 b(of)h(Optimalit)m(y)62 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)108 b(84)785 2688 y(4.3)94 b(Computational)29 b(Results)68 b(.)46 b(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(85)1085 2801 y(4.3.1)d(Cen)m(tral)30 b(Pro)5 b(jections)31 b(for)f(the)g Fq(nug)g Fw(Problems)88 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(87)1085 2914 y(4.3.2)d(Results)30 b(of)g(Sampling)e(the)j Fq(nug)e Fw(Problems)69 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.) 108 b(89)1085 3027 y(4.3.3)d(Computational)30 b(results)f(for)h(the)h Fq(lipa)e Fw(Problems)72 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)108 b(92)785 3140 y(4.4)94 b(Derandomization)e(.)46 b(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)g(.)f(.)h(.)108 b(95)785 3253 y(4.5)94 b(F)-8 b(urther)30 b(Questions)52 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(98)330 3457 y Fs(APPENDICES)35 b Fr(:)45 b(:)h(:)g(:)g(:)f(:)h(:)g(:) f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f (:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)c Fs(100)330 3660 y(BIBLIOGRAPHY)30 b Fr(:)46 b(:)g(:)f(:)h(:)g(:)f(:)h (:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)f(:)h(:) g(:)g(:)f(:)h(:)g(:)f(:)h(:)g(:)g(:)f(:)h(:)c Fs(110)2091 5680 y Fw(v)p eop %%Page: 6 7 6 6 bop 1658 504 a Ft(LIST)37 b(OF)h(T)-9 b(ABLES)330 1037 y Fs(T)g(able)p 330 1052 250 4 v 421 1263 a Fw(3.1)203 b(V)-8 b(alues)30 b(of)h(Extreme)g(Ra)m(ys)g(at)g Fr(")p Fw(,)g(on)f(2-cycles,)i(and)d(on)i(3-cycles)82 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(75)421 1489 y(4.1)203 b(V)-8 b(alues)30 b(of)h(P)m(arameter)h Fr(\013)1633 1503 y Fo(1)1703 1489 y Fw(on)e Fq(nug)f Fw(Problems)g(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)108 b(88)421 1715 y(4.2)203 b(T)-8 b(ransition)29 b(P)m(oin)m(ts)h(for)g Fr(\013)1653 1729 y Fo(1)1723 1715 y Fw(Compared)g(to)h Fq(nug)f Fw(P)m(arameters)73 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(89)421 1940 y(4.3)203 b(Standard)29 b(Deviation)i(of)f(Appro)m(ximation)f (Ratios)i(for)f(the)h Fq(nug)e Fw(Problems)57 b(.)46 b(.)f(.)h(.)108 b(90)421 2166 y(4.4)203 b(Best)31 b(Appro)m(ximation)e (Ratio)i(Obtained)e(in)g(Samples)g(for)h Fq(nug)g Fw(Problems)c(.)46 b(.)g(.)f(.)h(.)108 b(91)421 2392 y(4.5)203 b(F)-8 b(raction)31 b(of)g(P)m(erm)m(utations)f(ab)s(o)m(v)m(e)i Fr(\015)5 b(=h)p Fw(\()p Fr(n)p Fw(\))31 b(for)g Fq(nug)e Fw(Problems)e(.)45 b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(92)421 2618 y(4.6)203 b(T)-8 b(ransition)29 b(P)m(oin)m(ts)h(for)g Fr(\013)1653 2632 y Fo(1)1723 2618 y Fw(Compared)g(to)h Fq(lipa)e Fw(P)m(arameters)d(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h (.)108 b(93)421 2844 y(4.7)203 b(Best)31 b(Appro)m(ximation)e(Ratio)i (Obtained)e(in)g(Samples)g(for)h Fq(lipa)g Fw(Problems)49 b(.)d(.)f(.)h(.)108 b(93)421 3070 y(4.8)203 b(Standard)29 b(Deviation)i(of)f(Appro)m(ximation)f(Ratios)i(for)f(the)h Fq(lipa)e Fw(Problems)80 b(.)45 b(.)h(.)108 b(94)421 3295 y(4.9)203 b(F)-8 b(raction)31 b(of)g(P)m(erm)m(utations)f(ab)s(o)m (v)m(e)i Fr(\015)5 b(=h)p Fw(\()p Fr(n)p Fw(\))31 b(for)g Fq(lipa)e Fw(Problems)49 b(.)d(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(94)2078 5680 y(vi)p eop %%Page: 7 8 7 7 bop 1623 504 a Ft(LIST)37 b(OF)h(FIGURES)330 1037 y Fs(Figure)p 330 1069 297 4 v 421 1263 a Fw(2.1)203 b(Bullsey)m(e)30 b(Distribution)41 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)g(.)f(.)h(.)108 b(20)421 1489 y(2.2)203 b(Damp)s(ed)30 b(Oscillator)f(Distribution)44 b(.)h(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(23)421 1715 y(2.3)203 b(Spik)m(e)30 b(Distribution)86 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(27)421 1940 y(2.4)203 b(Sharp)29 b(Spik)m(e)57 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(31)421 2166 y(3.1)203 b(The)30 b(Base)h(of)g(the)f(Cen)m(tral)g(Cone) 49 b(.)d(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.) h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(53)421 2392 y(3.2)203 b(The)30 b(Cen)m(tral)g(\(Pure\))g(Cone)24 b(.)46 b(.)f(.)h(.)g(.)f(.)h (.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.) g(.)g(.)f(.)h(.)108 b(64)421 2618 y(3.3)203 b(The)30 b(Cen)m(tral)g(\(Symmetric\))g(Cone)68 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(69)421 2844 y(4.1)203 b(Diluted)29 b(Spik)m(e)h(Distribution)42 b(.)j(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g (.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(84)421 3070 y(4.2)203 b(F)-8 b(requency)31 b(of)f(Ratios)h Fr(f)1608 3084 y Fo(0)1647 3070 y Fw(\()p Fr(\033)s Fw(\))p Fr(=f)1862 3084 y Fo(0)1902 3070 y Fw(\()p Fr(\034)10 b Fw(\))31 b(for)g Fq(nug12)41 b Fw(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(90)421 3295 y(4.3)203 b(F)-8 b(requency)31 b(of)f(Ratios)h Fr(f)1608 3309 y Fo(0)1647 3295 y Fw(\()p Fr(\033)s Fw(\))p Fr(=f)1862 3309 y Fo(0)1902 3295 y Fw(\()p Fr(\034)10 b Fw(\))31 b(for)g Fq(nug30)41 b Fw(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g (.)f(.)h(.)g(.)g(.)f(.)h(.)108 b(91)2066 5680 y(vii)p eop %%Page: 1 9 1 8 bop 1793 728 a Ft(CHAPTER)35 b(I)1764 1057 y Fn(Bac)l(kground)471 1794 y Fv(In)c(discrete)g(optimization)d(problems,)i(an)h(ob)5 b(jectiv)m(e)31 b(function)g(is)f(giv)m(en)h(on)f(a)h(large)e(\014nite) 330 2010 y(set.)44 b(The)34 b(goal)d(is)h(to)g(\014nd)h(\\go)s(o)s(d")e (\(sa)m(y)j(particularly)c(large)h(or)i(small\))d(v)-5 b(alues)32 b(e\016cien)m(tly)-8 b(.)471 2227 y(Computational)41 b(complexit)m(y)h(theory)i(has)g(iden)m(ti\014ed)f(a)g(class)h(of)e (problems)h(whic)m(h)h(are)330 2444 y(called)32 b Fm(NP-c)-5 b(omplete)p Fv(.)46 b(The)34 b(NP-complete)f(problems)g(con)m(tain)g (man)m(y)g(of)g(the)h(com)m(binatorial)330 2661 y(problems)23 b(that)g(arise)g(in)g(practice,)i(suc)m(h)g(as)e(in)g(op)s(erations)g (researc)m(h,)k(but)c(seem)h(to)f(b)s(e)h(di\016cult)330 2877 y(to)k(solv)m(e)h(e\016cien)m(tly)-8 b(.)42 b(The)29 b(famous)f Fl(P)41 b Fv(=)27 b Fl(N)10 b(P)k Fv(?)57 b(question,)30 b(whic)m(h)e(asks)i(whether)f(there)g(is)f(an)m(y)330 3094 y(p)s(olynomial)19 b(time)j(algorithm)e(for)j(NP-complete)f (problems,)i(remains)e(the)i(cen)m(tral)f(problem)f(in)330 3311 y(computational)27 b(complexit)m(y)-8 b(.)41 b(Ho)m(w)m(ev)m(er,) 33 b(there)d(is)f(considerable)g(progress)h(in)e(understanding)330 3527 y(NP-complete)33 b(problems)g(b)m(y)i(using)e(to)s(ols)g(suc)m(h)i (as)f(appro)m(ximation,)e(randomness)i(and)g(non-)330 3744 y(p)s(olynomial)29 b(algorithms.)471 3961 y(The)e(goal)e(of)h (this)g(thesis)h(is)e(to)h(in)m(tro)s(duce)h(a)f(metho)s(d)f(of)h (studying)h(the)f(distribution)f(of)g(ob-)330 4177 y(jectiv)m(e)f(v)-5 b(alues)23 b(for)g(\(usually)f(NP-complete\))h(com)m(binatorial)c (optimization)h(problems)i(de\014ned)330 4394 y(on)35 b(groups.)50 b(This)35 b(approac)m(h)h(sho)m(ws)g(us)g(some)e(v)m(ery)j (general,)e(y)m(et)h(non-trivial)c(prop)s(erties)i(of)330 4611 y(the)f(optimization)28 b(function.)43 b(In)33 b(particular,)e(it) g(allo)m(ws)g(us)i(to)f(pro)s(duce)h(guaran)m(tees)g(for)f(sim-)330 4827 y(ple)40 b(p)s(olynomial)c(and)k(non-p)s(olynomial)c(appro)m (ximation)i(algorithms,)h(ev)-5 b(aluate)40 b(heuristics,)330 5044 y(and)33 b(helps)g(us)g(understand)h(wh)m(y)g(some)e(hard)h (problems)e(are)i(simpler)e(than)i(others.)471 5261 y(W)-8 b(e)26 b(fo)s(cus)h(on)e(t)m(w)m(o)i(imp)s(ortan)m(t)d(examples)i(of)f (com)m(binatorial)e(optimization)f(problems,)k(the)2092 5680 y Fw(1)p eop %%Page: 2 10 2 9 bop 2092 -210 a Fw(2)330 91 y Fv(Quadratic)39 b(Assignmen)m(t)h (Problem)f(\()p Fk(QAP)p Fv(\))h(and)g(the)h(T)-8 b(ra)m(v)m(eling)39 b(Salesman)g(Problem)g(\()p Fk(TSP)p Fv(\).)330 308 y(W)-8 b(e)33 b(obtain)e(results)i(on)g(b)s(oth)f(the)h Fm(fr)-5 b(e)g(quency)33 b Fv(and)f Fm(lo)-5 b(c)g(ation)32 b Fv(of)g(go)s(o)s(d)g(v)-5 b(alues.)471 525 y(Our)37 b(approac)m(h)g(is) f(to)g(use)i(tec)m(hniques)g(from)e(represen)m(tation)h(theory)g(\(see) h(Section)e(3.2\).)330 741 y(The)26 b(main)d(ob)5 b(ject)25 b(of)f(study)i(is)e(the)h(function)g(obtained)f(b)m(y)h(a)m(v)m (eraging)g(the)g(ob)5 b(jectiv)m(e)25 b(function)330 958 y(on)32 b(the)h(conjugacy)h(classes)f(of)f(p)s(erm)m(utations.)471 1175 y(This)k(Chapter)g(con)m(tains)f(bac)m(kground)i(material,)c (including)g(de\014nitions)i(and)g(some)h(his-)330 1391 y(tory)-8 b(.)41 b(W)-8 b(e)26 b(state)g(our)g(main)d(results)j(in)f (Chapter)i(I)s(I.)41 b(The)26 b(necessary)i(results)e(from)f(represen)m (ta-)330 1608 y(tion)j(theory)i(and)g(the)g(pro)s(ofs)f(are)g(in)g (Chapter)h(I)s(I)s(I.)42 b(Discussion)29 b(and)h(some)f(ancillary)e (results,)330 1825 y(including)41 b(computational)f(data)i(and)h (commen)m(ts)f(on)h(derandomization,)g(are)f(included)h(in)330 2042 y(Chapter)33 b(IV.)330 2308 y Ft(1.1)112 b(Problem)36 b(De\014nitions)471 2623 y Fv(Let)g(Mat)825 2638 y Fp(n)908 2623 y Fv(b)s(e)f(the)h(v)m(ector)h(space)g(of)e(all)e(real)i Fl(n)25 b Fu(\002)f Fl(n)36 b Fv(matrices)f Fl(A)e Fv(=)g(\()p Fl(a)3195 2638 y Fp(ij)3255 2623 y Fv(\),)k(1)32 b Fu(\024)i Fl(i;)17 b(j)38 b Fu(\024)c Fl(n)p Fv(.)330 2840 y(The)j(input)e(data)g (for)g(a)g Fk(QAP)i Fv(is)e(a)g(pair)g(of)g(matrices)g(in)g(Mat)2638 2855 y Fp(n)2685 2840 y Fv(;)i(w)m(e)f(refer)g(to)g(suc)m(h)h(a)e (problem)330 3057 y(as)f(ha)m(ving)f(size)h Fl(n)p Fv(.)46 b(T)-8 b(o)33 b(measure)h(the)g(complexit)m(y)f(of)g(an)g(algorithm,)e (w)m(e)j(coun)m(t)h(the)e(n)m(um)m(b)s(er)330 3273 y(of)42 b(arithmetic)e(op)s(erations)i(p)s(erformed)g(\(see)i(for)d(example)h ([AHU74]\).)74 b(This)42 b(is)g(the)h(usual)330 3490 y(mo)s(del)34 b(of)i(complexit)m(y)f(if)g(the)i(data)e(is)h(in)m (teger,)h(and)f(can)h(also)e(b)s(e)h(used)h(with)f(real)f(n)m(um)m(b)s (ers)330 3707 y([BCSS97)q(].)43 b(F)-8 b(rom)31 b(no)m(w)i(on,)g(w)m(e) h(assume)f(that)f Fl(n)c Fu(\025)g Fv(4.)471 3923 y(Let)f Fl(S)700 3938 y Fp(n)773 3923 y Fv(b)s(e)g(the)g(group)f(of)g(all)f(p)s (erm)m(utations)g Fl(\033)31 b Fv(of)26 b(the)h(set)g Fu(f)p Fv(1)p Fl(;)17 b(:)g(:)g(:)32 b(;)17 b(n)p Fu(g)p Fv(.)41 b(W)-8 b(e)27 b(are)g(in)m(terested)330 4140 y(in)40 b(the)h(action)f(of)h Fl(S)1109 4155 y Fp(n)1196 4140 y Fv(on)g(the)g(space)h(Mat)1961 4155 y Fp(n)2049 4140 y Fv(b)m(y)g(sim)m(ultaneous)e(p)s(erm)m(utations)g(of)g(ro)m(ws)i (and)330 4357 y(columns:)52 b(w)m(e)37 b(let)g Fl(\033)t Fv(\()p Fl(A)p Fv(\))e(=)g Fl(B)5 b Fv(,)38 b(where)g Fl(A)d Fv(=)g(\()p Fl(a)2142 4372 y Fp(ij)2203 4357 y Fv(\))i(and)g Fl(B)j Fv(=)35 b(\()p Fl(b)2776 4372 y Fp(ij)2837 4357 y Fv(\),)j(pro)m(vided)f Fl(b)3386 4372 y Fp(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))3677 4357 y Fv(=)e Fl(a)3839 4372 y Fp(ij)330 4573 y Fv(for)g(all)e Fl(i;)17 b(j)38 b Fv(=)32 b(1)p Fl(;)17 b(:)g(:)g(:)32 b(;)17 b(n)p Fv(.)51 b(Notice)35 b(that)g(\()p Fl(\033)t(\034)11 b Fv(\))p Fl(A)32 b Fv(=)g Fl(\033)t Fv(\()p Fl(\034)11 b(A)p Fv(\))36 b(for)f(an)m(y)h(t)m(w)m(o) f(p)s(erm)m(utations)g Fl(\033)k Fv(and)330 4790 y Fl(\034)11 b Fv(.)44 b(There)34 b(is)e(a)g(standard)h(scalar)f(pro)s(duct)h(on)f (Mat)2276 4805 y Fp(n)2323 4790 y Fv(:)555 5106 y Fu(h)p Fl(A;)17 b(B)5 b Fu(i)27 b Fv(=)h(trace\()p Fl(AB)1361 5065 y Fp(t)1391 5106 y Fv(\))g(=)1622 4982 y Fp(n)1571 5012 y Fj(X)1560 5222 y Fp(i;j)t Fo(=1)1743 5106 y Fl(a)1794 5121 y Fp(ij)1855 5106 y Fl(b)1896 5121 y Fp(ij)1957 5106 y Fl(;)114 b Fv(where)196 b Fl(A)28 b Fv(=)g(\()p Fl(a)2836 5121 y Fp(ij)2896 5106 y Fv(\))98 b(and)g Fl(B)32 b Fv(=)c(\()p Fl(b)3576 5121 y Fp(ij)3637 5106 y Fv(\))p eop %%Page: 3 11 3 10 bop 2092 -210 a Fw(3)330 91 y Ft(De\014nition)36 b(1.1.1.)49 b Fv(Let)23 b(us)f(\014x)h(t)m(w)m(o)g(matrices)e Fl(A)28 b Fv(=)g(\()p Fl(a)2418 106 y Fp(ij)2478 91 y Fv(\))22 b(and)h Fl(B)33 b Fv(=)27 b(\()p Fl(b)3007 106 y Fp(ij)3068 91 y Fv(\))22 b(and)g(let)g(us)h(consider)330 308 y(a)32 b(real-v)-5 b(alued)31 b(function)h Fl(f)39 b Fv(:)27 b Fl(S)1490 323 y Fp(n)1565 308 y Fu(\000)-16 b(!)27 b Fi(R)44 b Fv(de\014ned)34 b(b)m(y)943 624 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))27 b(=)h Fu(h)p Fl(B)5 b(;)17 b(\033)t Fv(\()p Fl(A)p Fv(\))p Fu(i)27 b Fv(=)1869 500 y Fp(n)1818 530 y Fj(X)1807 740 y Fp(i;j)t Fo(=1)1990 624 y Fl(b)2031 640 y Fp(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))2287 624 y Fl(a)2338 639 y Fp(ij)2427 624 y Fv(=)2592 500 y Fp(n)2541 530 y Fj(X)2530 740 y Fp(i;j)t Fo(=1)2713 624 y Fl(b)2754 639 y Fp(ij)2815 624 y Fl(a)2866 641 y Fp(\033)2908 622 y Fh(\000)p Fg(1)2991 641 y Fo(\()p Fp(i)p Fo(\))p Fp(\033)3111 622 y Fh(\000)p Fg(1)3196 641 y Fo(\()p Fp(j)t Fo(\))3699 624 y Fv(\(1.1\))330 941 y(The)36 b(problem)e(of)g(\014nding)h(a)g(p)s(erm)m(utation)e Fl(\033)39 b Fv(where)d(the)g(maxim)m(um)d(or)h(minim)m(um)e(v)-5 b(alue)34 b(of)330 1157 y Fl(f)45 b Fv(is)33 b(attained)g(is)h(kno)m (wn)h(as)f(the)h Fm(Quadr)-5 b(atic)36 b(Assignment)f(Pr)-5 b(oblem)33 b Fv(\()p Fk(QAP)p Fv(\).)i(It)f(is)g(one)g(of)f(the)330 1374 y(hardest)g(problems)f(of)g(Com)m(binatorial)e(Optimization.)471 1591 y(W)-8 b(e)32 b(sa)m(y)f(that)g(a)g Fk(QAP)h Fv(is)e Fm(symmetric)h Fv(if)f(the)h(matrix)e Fl(A)i Fv(is)g(symmetric.)42 b(If)31 b Fl(B)36 b Fv(is)31 b(symmetric,)330 1807 y(but)i Fl(A)f Fv(is)h(not,)f(then)h(w)m(e)h(can)f(switc)m(h)g(their)f(roles)g (to)g(get)h(a)f(symmetric)g(problem.)471 2099 y(The)i Fk(QAP)f Fv(is)f(a)g(sp)s(ecial)g(case)h(of)g(a)f(more)g(general)g (problem.)330 2390 y Ft(De\014nition)k(1.1.2.)49 b Fv(Supp)s(ose)37 b(w)m(e)f(are)f(giv)m(en)g(a)g(4-dimensional)d(arra)m(y)j(\(tensor\))h Fl(C)j Fv(=)3668 2280 y Fj(n)3734 2390 y Fl(c)3776 2343 y Fp(ij)3776 2418 y(k)r(l)3873 2390 y Fv(:)330 2607 y(1)27 b Fu(\024)i Fl(i;)17 b(j;)g(k)s(;)g(l)30 b Fu(\024)e Fl(n)993 2496 y Fj(o)1083 2607 y Fv(of)23 b Fl(n)1243 2571 y Fo(4)1307 2607 y Fv(real)f(n)m(um)m(b)s(ers.)42 b(The)24 b(general)f(problem)g(is)g(to)h(optimize)d(the)j(function)330 2824 y Fl(f)43 b Fv(is)32 b(de\014ned)i(b)m(y:)1712 3140 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))28 b(=)2099 3015 y Fp(n)2048 3045 y Fj(X)2037 3255 y Fp(i;j)t Fo(=1)2220 3140 y Fl(c)2262 3093 y Fp(ij)2262 3171 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))3699 3140 y Fv(\(1.2\))330 3477 y(If)40 b Fl(c)477 3430 y Fp(ij)477 3505 y(k)r(l)582 3477 y Fv(=)g Fl(a)749 3492 y Fp(ij)810 3477 y Fl(b)851 3492 y Fp(k)r(l)956 3477 y Fv(for)g(some)f(matrices)h Fl(A)g Fv(=)h(\()p Fl(a)2082 3492 y Fp(ij)2142 3477 y Fv(\))f(and)g Fl(B)46 b Fv(=)40 b(\()p Fl(b)2732 3492 y Fp(k)r(l)2797 3477 y Fv(\),)i(w)m(e)f(get)f(the)h(sp)s(ecial)e(case) 330 3694 y(\(1.1\))d(w)m(e)h(started)f(with.)54 b(The)37 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y(assigning)43 b Fl(n)h Fv(facilities)d(to)i Fl(n)h Fv(lo)s(cations)e (so)i(as)g(to)g(minimize)c(the)k(total)f(cost.)77 b(Costs)45 b(arise)p eop %%Page: 4 12 4 11 bop 2092 -210 a Fw(4)330 91 y Fv(from)38 b(transp)s(orting)g (materials)f(b)s(et)m(w)m(een)k(facilities;)f(they)g(are)g(prop)s (ortional)c(to)j(the)h(\\\015o)m(w")330 308 y(of)34 b(go)s(o)s(ds)g Fl(a)773 323 y Fp(ij)869 308 y Fv(required)h(from)f(facilit)m(y)e Fl(i)j Fv(to)g(facilit)m(y)e Fl(j)40 b Fv(and)35 b(the)g(distances)h Fl(b)3229 323 y Fp(k)r(l)3329 308 y Fv(from)d(lo)s(cation)330 525 y Fl(k)44 b Fv(to)c(lo)s(cation)e Fl(l)r Fv(.)68 b(Then)42 b(for)e(a)g(giv)m(en)h(assignmen)m(t)g Fl(\033)j Fv(of)c(facilities)e(to)i(lo)s(cations,)h(the)g(total)330 741 y(transp)s(ortation)33 b(cost)i(is)g(the)g(sum)f(of)g(these)i (costs)g(o)m(v)m(er)g(all)c(pairs)i(of)g(facilities,)f(equal)h(to)g (the)330 958 y(ob)5 b(jectiv)m(e)34 b(function)e(\(1.1\).)471 1175 y(Ko)s(opmans)27 b(and)i(Bec)m(kmann)g(also)e(consider)i(a)f 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(whic)m(h)e(is)f(exp)s(ensiv)m(e)j(to)d(store)330 3758 y(and)27 b(manipulate.)39 b(Since)27 b(the)g(general)f(problem)f(has)i (not)f(sho)m(wn)i(a)e(substan)m(tial)g(adv)-5 b(an)m(tage)27 b(in)330 3974 y(mo)s(deling)h(practical)i(problems,)g(most)h(w)m(ork)h (is)e(done)h(on)g(the)h(sp)s(ecial)e(cases)i(\(1.1\))e(and)h(\(1.3\),) 330 4191 y(whic)m(h)i(require)g(only)f Fl(O)s Fv(\()p Fl(n)1325 4155 y Fo(2)1364 4191 y Fv(\))g(storage.)330 4482 y Ft(De\014nition)k(1.1.3.)49 b Fv(One)31 b(of)g(the)g(most)f (prominen)m(t)g(problems)g(in)g(computational)e(complex-)330 4699 y(it)m(y)d(is)g(the)g Fm(T)-7 b(r)i(aveling)27 b(Salesman)f(Pr)-5 b(oblem)24 b Fv(\()p Fk(TSP)p Fv(\).)i(F)-8 b(or)25 b(com)m(binatorial) c(optimization,)j(the)h Fk(TSP)330 4916 y Fv(is)37 b(considered)i(as)f (an)f Fm(optimization)g Fv(problem)f(on)i(the)g(the)g(complete)f(graph) g(on)h Fl(n)g Fv(v)m(ertices,)330 5132 y Fl(K)413 5147 y Fp(n)460 5132 y Fv(.)44 b(A)33 b(w)m(eigh)m(t)g(is)f(assigned)h(to)f 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Fv(1)97 b(if)64 b Fl(j)28 b Fu(\000)23 b Fl(i)28 b Fv(=)f(1)100 b(mo)s(d)32 b Fl(n)2582 4169 y Fv(0)97 b(otherwise)330 4800 y(Let)33 b Fl(B)k Fv(the)c(con)m(tain)g(the)g(\(asymmetric\))e(w)m(eigh)m(ts:) 1566 5090 y Fl(b)1607 5105 y Fp(ij)1695 5090 y Fv(=)d(W)-8 b(eigh)m(t)32 b(of)g(edge)h(\()p Fl(i;)17 b(j)6 b Fv(\))330 5380 y(Then)34 b Fu(h)p Fl(B)5 b(;)17 b(\033)t Fv(\()p Fl(A)p Fv(\))p Fu(i)32 b Fv(giv)m(es)h(the)g(w)m(eigh)m(t)f(of)h(cycle) g Fl(\033)t Fv(.)p eop %%Page: 6 14 6 13 bop 2092 -210 a Fw(6)330 91 y Ft(De\014nition)36 b(1.1.4.)49 b Fv(It)43 b(is)f(in)m(teresting)g(to)g(compare)g(the)g Fk(QAP)h Fv(to)f(the)h Fm(Line)-5 b(ar)44 b(Assignment)330 308 y(Pr)-5 b(oblem)32 b Fv(\()p Fk(LAP)p Fv(\))h(of)f(maximizing:)1772 623 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))27 b(=)2147 499 y Fp(n)2097 529 y Fj(X)2112 739 y Fp(i)p Fo(=1)2257 623 y Fl(d)2308 639 y Fp(i\033)r Fo(\()p Fp(i)p Fo(\))3699 623 y Fv(\(1.4\))330 938 y(This)i(mo)s(dels)f(the)h(problem)f(of)g (assigning)g Fl(n)h 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Fp(n)1742 5193 y Fj(X)1731 5403 y Fp(i;j)t Fo(=1)1914 5287 y Fl(c)1956 5240 y Fp(ij)1956 5319 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))2239 5287 y Fv(=)2405 5163 y Fp(n)2354 5193 y Fj(X)2343 5403 y Fp(i;j)t Fo(=1)2526 5287 y Fl(c)2568 5240 y Fo(1)p Fp(j)2568 5319 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))p eop %%Page: 7 15 7 14 bop 2092 -210 a Fw(7)330 91 y Fv(T)-8 b(o)42 b(reduce)i(to)d(the)i (linear)d(assignmen)m(t)i(problem,)i(w)m(e)f(tak)m(e)g(the)f(matrix)f Fl(D)46 b Fv(=)e(\()p Fl(d)3533 106 y Fp(j)t(l)3591 91 y Fv(\))e(to)g(b)s(e)330 308 y(de\014ned)34 b(b)m(y:)1858 525 y Fl(d)1909 540 y Fp(j)t(l)1994 525 y Fv(=)2149 400 y Fp(n)2098 430 y Fj(X)2106 642 y Fp(k)r Fo(=1)2258 525 y Fl(c)2300 477 y Fo(1)p Fp(j)2300 552 y(k)r(l)471 795 y Fv(No)m(w)k(if)e Fl(A)h Fv(in)f(\(1.1\))h(has)g(constan)m(t)h(ro)m (ws,)i(then)d Fl(\033)42 b Fv(acts)37 b(on)g Fl(A)g Fv(b)m(y)h(p)s(erm) m(uting)e(ro)m(ws.)58 b(W)-8 b(e)330 1012 y(reduce)34 b(to)e(\(1.4\))g(b)m(y)i(taking)d Fl(D)g Fv(=)c(\()p Fl(d)1731 1027 y Fp(ik)1798 1012 y Fv(\))32 b(giv)m(en)h(b)m(y:)1800 1328 y Fl(d)1851 1343 y Fp(ik)1945 1328 y Fv(=)2099 1204 y Fp(n)2049 1234 y Fj(X)2065 1446 y Fp(l)q Fo(=1)2209 1328 y Fl(a)2260 1343 y Fp(i)p Fo(1)2324 1328 y Fl(b)2365 1343 y Fp(k)r(l)330 1645 y Fv(F)-8 b(or)36 b(the)h(generalized)f (problem)f(\(1.2\),)i(if)f(for)g(all)e Fl(k)40 b Fv(and)c Fl(l)r Fv(,)i(the)f(matrix)e Fl(A)g Fv(=)f(\()p Fl(a)3403 1660 y Fp(ij)3464 1645 y Fv(\))i(giv)m(en)h(b)m(y)330 1861 y Fl(a)381 1876 y Fp(ij)469 1861 y Fv(=)28 b Fl(c)615 1814 y Fp(ij)615 1889 y(k)r(l)712 1861 y Fv(has)33 b(constan)m(t)g(ro)m (ws,)h(w)m(e)g(reduce)f(to)g(the)g Fk(LAP)g Fv(b)m(y)h(de\014ning)e Fl(D)e Fv(=)e(\()p Fl(d)3266 1876 y Fp(ik)3332 1861 y Fv(\))33 b(b)m(y:)1857 2178 y Fl(d)1908 2193 y Fp(ik)2002 2178 y Fv(=)2156 2053 y Fp(n)2106 2083 y Fj(X)2122 2295 y Fp(l)q Fo(=1)2266 2178 y Fl(c)2308 2137 y Fp(i)p Fo(1)2308 2202 y Fp(k)r(l)471 2494 y Fv(Finally)-8 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5347 y(b)m(y)33 b(the)f(tensor)g Fl(C)j Fv(=)27 b(\()p Fl(c)1212 5300 y Fp(ij)1212 5375 y(k)r(l)1277 5347 y Fv(\),)32 b(w)m(e)g(write)g Fl(C)i Fv(=)28 b Fl(C)2043 5362 y Fo(1)2103 5347 y Fv(+)20 b Fl(C)2269 5362 y Fo(2)2308 5347 y Fv(,)32 b(where)h Fl(C)2718 5362 y Fo(1)2789 5347 y Fv(is)e(the)h(pro)5 b(jection)32 b(of)f Fl(C)39 b Fv(in)m(to)p eop %%Page: 8 16 8 15 bop 2092 -210 a Fw(8)330 91 y Fv(the)34 b(v)m(ector)g(space)g (spanned)g(b)m(y)g(tensors)g(where)h(for)d(all)f Fl(k)s(;)17 b(l)35 b Fv(the)f(matrices)e Fl(A)d Fv(=)f(\()p Fl(a)3486 106 y Fp(ij)3547 91 y Fv(\))g(=)h(\()p Fl(c)3798 44 y Fp(ij)3798 119 y(k)r(l)3862 91 y Fv(\))330 308 y(ha)m(v)m(e)43 b(constan)m(t)f(columns,)h(tensors)g(where)f(all)d(suc)m(h)k Fl(A)f Fv(ha)m(v)m(e)g(constan)m(t)h(ro)m(ws,)h(and)e(tensors)330 525 y(where)36 b(all)c(suc)m(h)k Fl(A)f Fv(are)f(diagonal.)47 b(W)-8 b(e)35 b(call)e Fl(C)2100 540 y Fo(1)2174 525 y Fv(the)i Fm(line)-5 b(ar)34 b Fv(part)g(of)g(the)h(problem,)f(and)h (note)330 741 y(that)41 b(if)g Fl(C)50 b Fv(=)43 b Fl(C)958 756 y Fo(1)997 741 y Fv(,)h(then)f(the)f(problem)e(\(1.2\))h(reduces)j (to)d(the)h Fk(LAP)p Fv(.)h(W)-8 b(e)42 b(giv)m(e)f(form)m(ulas)f(for) 330 958 y(computing)31 b Fl(A)885 973 y Fo(1)957 958 y Fv(\(or)i Fl(C)1185 973 y Fo(1)1224 958 y Fv(\))f(giv)m(en)h Fl(A)g Fv(\(or)f Fl(C)7 b Fv(\))32 b(in)g(Section)g(3.2.3.)471 1175 y(The)i(remaining)d(comp)s(onen)m(t)i(\()p Fl(A)1738 1190 y Fo(2)1778 1175 y Fl(;)17 b(B)5 b Fv(\))32 b(for)h(a)g(problem)f (of)g(t)m(yp)s(e)i(\(1.1\))f(de\014nes)i(a)e Fk(QAP)p Fv(.)g(W)-8 b(e)330 1391 y(call)37 b(this)i(the)h Fm(pur)-5 b(e)39 b Fv(comp)s(onen)m(t)g(of)f(the)i Fk(QAP)p Fv(.)g(In)f (particular)e(w)m(e)j(call)e(a)g Fk(QAP)i Fm(pur)-5 b(e)39 b Fv(if)f(in)g(the)330 1608 y(decomp)s(osition)27 b(ab)s(o)m(v)m(e)i(w) m(e)g(ha)m(v)m(e)h Fl(A)e Fv(=)g Fl(A)1879 1623 y Fo(2)1918 1608 y Fv(.)42 b(W)-8 b(e)29 b(note)g(that)f Fl(A)2644 1623 y Fo(2)2712 1608 y Fv(m)m(ust)h(ha)m(v)m(e)h(constan)m(t)f(ro)m(w) g(and)330 1825 y(column)j(sums,)h(and)g(a)f(constan)m(t)i(diagonal.)41 b(Similarly)-8 b(,)29 b(for)j(the)i(generalized)e(problem)f(\(1.2\),) 330 2042 y(the)i(comp)s(onen)m(t)g Fl(C)1067 2057 y Fo(2)1139 2042 y Fv(de\014nes)h(the)f Fm(pur)-5 b(e)33 b Fv(comp)s(onen)m(t)g(of) f(the)h Fk(QAP)p Fv(,)h(and)e(w)m(e)i(call)d(a)h(generalized)330 2258 y(problem)44 b Fm(pur)-5 b(e)45 b Fv(if)f(in)h(the)g(decomp)s (osition)e(w)m(e)k(ha)m(v)m(e)f Fl(C)56 b Fv(=)49 b Fl(C)2729 2273 y Fo(2)2768 2258 y Fv(.)82 b(F)-8 b(or)44 b(an)m(y)i Fl(k)s(;)17 b(l)47 b Fv(the)e(matrix)330 2475 y Fl(A)54 b Fv(=)f(\()p Fl(A)697 2490 y Fp(ij)758 2475 y Fv(\))h(=)f(\()p Fl(C)1087 2490 y Fo(2)1126 2475 y Fv(\))1164 2428 y Fp(ij)1164 2503 y(k)r(l)1277 2475 y Fv(m)m(ust)48 b(ha)m(v)m(e)h(constan)m(t)f(ro) m(w)h(and)f(column)e(sums,)52 b(and)c(a)g(constan)m(t)330 2692 y(diagonal.)330 2951 y Ft(De\014nition)36 b(1.1.5.)49 b Fv(W)-8 b(e)41 b(in)m(tro)s(duce)g(the)g(standard)g(Hamming)d(metric) h(on)i(the)g(symmetric)330 3168 y(group)32 b Fl(S)666 3183 y Fp(n)713 3168 y Fv(.)43 b(F)-8 b(or)31 b(t)m(w)m(o)i(p)s(erm)m (utations)e Fl(\034)6 b(;)17 b(\033)31 b Fu(2)d Fl(S)2069 3183 y Fp(n)2116 3168 y Fv(,)k(let)f(the)i(distance)f(dist\()p Fl(\033)n(;)17 b(\034)11 b Fv(\))32 b(b)s(e)g(the)g(n)m(um)m(b)s(er)330 3384 y(of)g(indices)g(1)c Fu(\024)g Fl(i)g Fu(\024)g Fl(n)33 b Fv(where)h Fl(\033)i Fv(and)d Fl(\034)44 b Fv(disagree:)1527 3656 y(dist\()p Fl(\033)n(;)17 b(\034)11 b Fv(\))28 b(=)g Fu(j)p Fl(i)f Fv(:)h Fl(\033)t Fv(\()p Fl(i)p Fv(\))g Fu(6)p Fv(=)f Fl(\034)11 b Fv(\()p Fl(i)p Fv(\))p Fu(j)p Fl(:)330 3927 y Fv(One)30 b(can)g(observ)m(e)h(that)e (the)h(distance)g(is)f(in)m(v)-5 b(arian)m(t)28 b(under)j(the)f(left)e (and)i(righ)m(t)f(actions)g(of)g Fl(S)3826 3942 y Fp(n)3873 3927 y Fv(:)1168 4199 y(dist\()p Fl(\033)t(\033)1478 4214 y Fo(1)1517 4199 y Fl(;)17 b(\033)t(\033)1675 4214 y Fo(2)1715 4199 y Fv(\))28 b(=)f(dist\()p Fl(\033)2135 4214 y Fo(1)2175 4199 y Fl(;)17 b(\033)2274 4214 y Fo(2)2313 4199 y Fv(\))28 b(=)f(dist\()p Fl(\033)2733 4214 y Fo(1)2773 4199 y Fl(\033)n(;)17 b(\033)2925 4214 y Fo(2)2965 4199 y Fl(\033)t Fv(\))330 4470 y(for)32 b(all)f Fl(\033)670 4485 y Fo(1)709 4470 y Fl(;)17 b(\033)808 4485 y Fo(2)848 4470 y Fl(;)g(\033)31 b Fu(2)d Fl(S)1132 4485 y Fp(n)1179 4470 y Fv(.)471 4730 y(If)f(w)m(e)h(are)e(mo)s(delling)e(a)i(problem)g (using)g(the)h Fk(QAP)h Fv(\(suc)m(h)g(as)f(the)g(facilit)m(y)e(lo)s (cation)f(example)330 4946 y(in)m(tro)s(duced)k(in)g(Section)f (1.1.1\),)i(then)g(p)s(erm)m(utations)e(are)h(nearb)m(y)h(in)e(the)i (Hamming)d(distance)330 5163 y(if)j(they)i(in)m(v)m(olv)m(e)f(making)f (man)m(y)h(of)f(the)i(same)e(assignmen)m(ts.)43 b(The)31 b(Hamming)d(distance)i(is)g(ex-)330 5380 y(actly)d(the)h(n)m(um)m(b)s (er)f(of)g(assignmen)m(ts)g(that)g(di\013er)g(b)s(et)m(w)m(een)i(the)f (p)s(erm)m(utations.)41 b(In)28 b(particular,)p eop %%Page: 9 17 9 16 bop 2092 -210 a Fw(9)330 91 y Fv(the)33 b(distance)f(do)s(es)h (not)g(dep)s(end)g(on)f(the)h(c)m(hoice)g(of)f(the)h(n)m(um)m(b)s (ering)e(of)h(the)h(plan)m(ts)f(from)f(the)330 308 y(set)d Fu(f)p Fv(1)p Fl(;)17 b Fv(2)p Fl(;)g(:::;)g(n)p Fu(g)p Fv(,)28 b(since)g(the)g(distance)g(is)f(in)m(v)-5 b(arian)m(t)26 b(under)i(conjugation)f(b)m(y)h(an)m(y)g(p)s(erm)m(utation)330 525 y Fl(!)t Fv(:)1439 741 y(dist\()p Fl(\033)n(;)17 b(\034)11 b Fv(\))28 b(=)g(dist\()p Fl(!)t(\033)t(!)2340 700 y Ff(\000)p Fo(1)2433 741 y Fl(;)17 b(!)t(\034)11 b(!)2660 700 y Ff(\000)p Fo(1)2753 741 y Fv(\))330 1012 y(In)33 b(this)f(case,)i Fl(!)i Fv(p)s(erm)m(utes)d(the)g(lab)s(els)e (of)h(the)h(plan)m(ts.)471 1229 y(The)i(notion)f(of)g(\\nearness")h (measured)g(b)m(y)h(the)e(Hamming)e(distance)j(is)f(appropriate)f(for) 330 1445 y(the)27 b(con)m(text)g(of)f(a)g(heuristic)g(\\lo)s(cal)d (searc)m(h")28 b(approac)m(h)e(to)g(solving)f(an)h(optimization)d (problem)330 1662 y(on)32 b(p)s(erm)m(utations.)43 b(W)-8 b(e)33 b(discuss)h(heuristic)e(approac)m(hes)i(to)e(the)h Fk(QAP)g Fv(in)f(Section)h(4.2.)471 1879 y(There)44 b(are)f(sev)m(eral) h(other)f(metrics)f(on)h Fl(S)2090 1894 y Fp(n)2137 1879 y Fv(,)i(see)f(for)e(example)h(c)m(hapter)h(6B)e(of)h([Dia88)n(].)330 2095 y(Most)g(do)g(not)g(mak)m(e)g(sense)h(for)f(our)g(problem)e(\(for) h(example,)j(they)f(c)m(hange)g(substan)m(tially)330 2312 y(under)d(relab)s(eling\).)64 b(One)40 b(metric)f(that)h(is)g (quite)g(similar)d(to)j(the)g(Hamming)e(metric)h(is)g(the)330 2529 y Fm(Cayley)h(metric)e Fv(of)g(the)h(\(generating\))f(set)h(of)f (all)e(transp)s(ositions.)60 b(This)39 b(metric)e(coun)m(ts)i(the)330 2746 y(minim)m(um)26 b(n)m(um)m(b)s(er)k(of)g(2-cycles)g(\(transp)s (ositions\))e(required)i(to)f(c)m(hange)i Fl(\034)41 b Fv(to)29 b Fl(\033)t Fv(.)43 b(Our)29 b(results)330 2962 y(w)m(ould)k(not)f(lo)s(ok)f(to)s(o)h(di\013eren)m(t)h(if)e(w)m(e) j(used)g(the)f(Ca)m(yley)g(metric.)330 3254 y Ft(De\014nition)j(1.1.6.) 49 b Fv(F)-8 b(or)30 b(a)g(p)s(erm)m(utation)f Fl(\034)42 b Fv(and)30 b(an)g(in)m(teger)h Fl(k)f(>)e Fv(1,)j(w)m(e)g(consider)g (the)f(\\)p Fl(k)s Fv(-th)330 3470 y(ring")h(around)i Fl(\034)11 b Fv(:)1269 3687 y Fl(U)f Fv(\()p Fl(\034)c(;)17 b(k)s Fv(\))28 b(=)1698 3606 y Fj(\010)1756 3687 y Fl(\033)k Fu(2)c Fl(S)1997 3702 y Fp(n)2072 3687 y Fv(:)f(dist\()p Fl(\033)n(;)17 b(\034)11 b Fv(\))29 b(=)e Fl(n)22 b Fu(\000)h Fl(k)2876 3606 y Fj(\011)2934 3687 y Fl(:)330 3958 y Fv(Hence)39 b(for)d(an)m(y)i(p)s(erm)m(utation)e Fl(\034)49 b Fv(the)38 b(group)f Fl(S)2135 3973 y Fp(n)2219 3958 y Fv(splits)f(in)m(to)h(the)g(disjoin)m(t)g(union)f(of)h Fl(n)g Fv(rings)330 4174 y Fl(U)10 b Fv(\()p Fl(\034)c(;)17 b(k)s Fv(\))33 b(for)f Fl(k)f Fv(=)c(0)p Fl(;)17 b Fv(1)p Fl(;)g Fv(2)p Fl(;)g(:)g(:)g(:)31 b(;)17 b(n)22 b Fu(\000)h Fv(2)p Fl(;)17 b(n)p Fv(.)471 4466 y(The)34 b(innermost)d(ring)h Fl(U)10 b Fv(\()p Fl(\034)c(;)17 b(n)p Fv(\))32 b(con)m(tains)h(the)g (single)e(p)s(erm)m(utation)g Fl(\034)11 b Fv(,)33 b(and)g(the)g(next)g (ring)330 4682 y Fl(U)10 b Fv(\()p Fl(\034)c(;)17 b(n)8 b Fu(\000)g Fv(2\))26 b(has)967 4542 y Fj(\022)1041 4615 y Fl(n)1045 4751 y Fv(2)1099 4542 y Fj(\023)1198 4682 y Fv(p)s(erm)m(utations)f(that)g(di\013er)g(from)f Fl(\034)38 b Fv(b)m(y)26 b(a)f(sw)m(apping)h(a)g(pair)e(of)i(elemen)m(ts)330 4899 y(\(that)h(is,)h(b)m(y)h(a)e(2-cycle\).)41 b(The)29 b(size)f(of)e(the)i(ring)f Fl(U)10 b Fv(\()p Fl(\034)c(;)17 b(k)s Fv(\))28 b(increases)g(as)g Fl(k)i Fv(decreases,)h(and)c(most)330 5116 y(of)k(the)h(p)s(erm)m(utations)f(are)g(con)m(tained)h(in)e(the)i (outermost)f(rings.)43 b(In)32 b(Lemma)e(3.4.2,)h(w)m(e)i(sho)m(w)330 5333 y(that)44 b(the)g(n)m(um)m(b)s(er)g(of)g(p)s(erm)m(utations)f(in)g (the)h(outermost)g(ring)f Fl(U)10 b Fv(\()p Fl(\034)c(;)17 b Fv(0\))44 b(is)f(at)h(least)3644 5265 y Fl(n)p Fv(!)p 3644 5310 86 4 v 3663 5401 a(3)3783 5333 y(for)p eop %%Page: 10 18 10 17 bop 2070 -210 a Fw(10)330 91 y Fl(n)37 b Fu(\025)h Fv(2.)59 b(Using)38 b(this)g(fact,)h(it)e(is)h(easy)h(to)e(sho)m(w)j 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1467 59 4 v 22 w Fl(f)330 1865 y Fv(b)s(e)40 b(the)h(\\shifted")e(function.)65 b(Hence)42 b(the)e(a)m(v)m(erage)h(v)-5 b(alue)39 b(of)h Fl(f)2766 1880 y Fo(0)2845 1865 y Fv(is)g(0.)65 b(Let)40 b Fl(\034)52 b Fv(b)s(e)40 b(a)f(p)s(erm)m(u-)330 2081 y(tation)d(where)i(the)f(maxim)m(um)e(v)-5 b(alue)36 b(of)h Fl(f)1960 2096 y Fo(0)2036 2081 y Fv(is)g(attained,)g(so)g Fl(f)2728 2096 y Fo(0)2768 2081 y Fv(\()p Fl(\034)11 b Fv(\))35 b Fu(\025)h Fl(f)3093 2096 y Fo(0)3132 2081 y Fv(\()p Fl(\033)t Fv(\))h(for)f(all)f Fl(\033)k Fu(2)d Fl(S)3853 2096 y Fp(n)330 2298 y Fv(and)41 b Fl(f)576 2313 y Fo(0)616 2298 y Fv(\()p Fl(\034)11 b Fv(\))42 b Fl(>)g Fv(0)e(unless)i Fl(f)1339 2313 y Fo(0)1420 2298 y Fu(\021)h Fv(0)d(\(the)i(problem)d(with)i(minim)m(um)c(instead)k(of)g (maxim)m(um)d(is)330 2515 y(completely)31 b(similar\).)471 2806 y(W)-8 b(e)33 b(remark)f(that)h(it)e(is)h(easy)i(to)e(compute)h (the)g(a)m(v)m(erage)g(v)-5 b(alue)p 2878 2725 V 32 w Fl(f)44 b Fv(\(see)33 b(Lemma)e(3.1.1\).)330 3058 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3870 y(\014)2048 3895 y Fl(i)g Fv(:)g Fl(\033)t Fv(\()p Fl(i)p Fv(\))g(=)f Fl(i)2496 3810 y Fj(\014)2496 3870 y(\014)330 4153 y Fv(and)33 b(let)f Fl(t)g Fv(b)s(e)h(the)g(n)m(um)m(b)s(er)g(of)f Fm(tr)-5 b(ansp)g(ositions)32 b Fv(\(or)g(2-cycles\))h(in)f Fl(\033)t Fv(:)1263 4442 y Fl(t)p Fv(\()p Fl(\033)t Fv(\))c(=)1564 4357 y Fj(\014)1564 4417 y(\014)1598 4442 y Fl(i)g(<)f(j)34 b Fv(:)28 b Fl(\033)t Fv(\()p Fl(i)p Fv(\))f(=)h Fl(j)104 b Fv(and)97 b Fl(\033)t Fv(\()p Fl(j)6 b Fv(\))28 b(=)f Fl(i)2933 4357 y Fj(\014)2933 4417 y(\014)471 4730 y Fv(One)47 b(can)f(sho)m(w)h(that)f Fl(p)p Fv(\()p Fl(\033)t Fv(\),)j Fl(p)1672 4694 y Fo(2)1711 4730 y Fv(\()p Fl(\033)t Fv(\))d(and)g Fl(t)p Fv(\()p Fl(\033)t Fv(\))h(are)f(functions)g(of)f (t)m(yp)s(e)i(\(1.2\))f(for)f(some)330 4946 y(particular)31 b(tensors)1111 4866 y Fj(\010)1170 4946 y Fl(c)1212 4899 y Fp(ij)1212 4974 y(k)r(l)1276 4866 y Fj(\011)1334 4946 y Fv(,)i(see)h(Remark)e(3.2.5.)330 5380 y(In)k(the)g(next)h(t)m(w)m(o)f 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(ideas)h(in)g(complexit)m(y)f(theory)i(of)f(reductions)h(and)g (completeness,)h(w)m(ere)330 1973 y(originally)28 b(used)33 b(\(for)e(example)g(in)g(Karp's)g(pap)s(er\))h(in)f(the)h(con)m(text)h (of)e(\\decision)g(problems",)330 2190 y(that)24 b(is)f(problems)g (with)g(a)h(`y)m(es')i(or)d(`no')h(answ)m(er.)42 b(The)25 b Fk(TSP)f Fm(de)-5 b(cision)23 b Fv(problem)f(is)i(to)f(determine)330 2407 y(if)h(a)g(giv)m(en)h(graph)f(has)i(an)m(y)f(Hamiltonian)c(cycle.) 42 b(W)-8 b(e)25 b(are)f(in)m(terested)i(in)e(the)h Fk(TSP)h Fm(optimization)330 2623 y Fv(problem)32 b(of)h(\014nding)f(a)h(cycle)h (of)f(maxim)m(um)d(\(or)j(minim)m(um\))d(w)m(eigh)m(t.)46 b(F)-8 b(or)32 b(the)h(optimization)330 2840 y(problem,)e(w)m(e)j(can)f (assume)g(that)f(w)m(eigh)m(ts)i(are)e(giv)m(en)h(on)g(the)g(complete)e (graph,)i Fl(K)3474 2855 y Fp(n)3521 2840 y Fv(.)330 3092 y Fs(1.2.1)105 b(Complexit)m(y)34 b(of)h(Optimization)f(Problems) 471 3373 y Fv(The)40 b(ideas)f(of)g(reductions)g(and)g(completeness)h 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y(ratio)i(is)h(meaningful,)g(w)m(e)h(can)g(not)f(allo)m(w)f(arbitrary)g (edge)i(w)m(eigh)m(ts,)i(b)s(ecause)f(the)f(optimal)330 958 y(v)-5 b(alue)32 b(could)g(b)s(e)h(zero)g(\(for)f(example\).)471 1175 y(One)45 b(w)m(a)m(y)h(to)f(\014x)h(this)e(is)g(to)h(assume)g (that)g(all)e(the)i(w)m(eigh)m(ts)h(are)f(p)s(ositiv)m(e.)79 b(Then)46 b(the)330 1391 y Fk(TSP)34 b Fm(minimization)e Fv(problem)g(is)h Fm(NPO-c)-5 b(omplete)p Fv(,)32 b(that)i(is)e(an)m(y) i(p)s(olynomial)c(time)i(algorithm)330 1608 y(guaran)m(teeing)22 b(a)h(v)-5 b(alue)22 b Fl(\033)27 b Fv(with)1516 1541 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))p 1516 1585 V 1519 1676 a Fl(f)g Fv(\()p Fl(\034)g Fv(\))1747 1608 y Fu(\024)28 b Fv(2)1901 1567 y Ff(\000)p Fp(n)1999 1544 y Fe(\024)2043 1608 y Fv(,)d(for)d(some)h Fl(\024)28 b(>)f Fv(0)c(implies)d Fl(P)41 b Fv(=)28 b Fl(N)10 b(P)37 b Fv([A)m(CG)3689 1572 y Fo(+)3748 1608 y Fv(99].)330 1825 y(This)c(is)f(the)h(strongest) g(t)m(yp)s(e)h(of)e(non-appro)m(ximabilit)m(y)d(result.)471 2042 y(In)40 b(con)m(trast,)j(the)d(\(asymmetric\))f Fk(TSP)i Fm(maximization)d Fv(problem)h(with)h(p)s(ositiv)m(e)f(w)m (eigh)m(ts)330 2258 y(admits)34 b(a)h(p)s(olynomial)d(time)i(constan)m (t)i(appro)m(ximation)d(guaran)m(teeing)3108 2191 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))p 3108 2235 V 3111 2327 a Fl(f)g Fv(\()p Fl(\034)g Fv(\))3344 2258 y Fu(\025)3488 2191 y Fv(8)p 3464 2235 98 4 v 3464 2327 a(13)3572 2258 y(\(1)23 b Fu(\000)3799 2191 y Fv(1)p 3794 2235 59 4 v 3794 2327 a Fl(n)3862 2258 y Fv(\))330 2475 y(where)45 b Fl(n)e Fv(is)g(the)h(size)g(of)f(the)h(problem)f([Bl\177)-49 b(a02)n(].)76 b(If)44 b(the)g Fk(TSP)g Fv(is)f(symmetric,)j(this)d(can) h(b)s(e)330 2692 y(impro)m(v)m(ed)23 b(to)g(3)p Fl(=)p Fv(4)f([Ser84],)j(and)e(there)h(is)e(a)h(randomized)f(algorithm)e(whic) m(h)k(returns)g(a)e(solution)330 2908 y(with)32 b(exp)s(ected)j(ratio)c Fl(r)36 b Fv(for)c(an)m(y)h(ratio)f Fl(r)e(<)e Fv(25)p Fl(=)p Fv(33)k([HR00)o(].)44 b(Rep)s(ortedly)-8 b(,)33 b(these)h(ratios)e(ha)m(v)m(e)330 3125 y(recen)m(tly)27 b(b)s(een)g(impro)m(v)m(ed)g(to)f(5)p Fl(=)p Fv(8)f([LS)q(])h(in)g(the) g(asymmetric)g(case,)i(and)f(7)p Fl(=)p Fv(8)e(in)h(the)h(symmetric)330 3342 y(case.)330 3594 y Fs(1.2.2)105 b(Appro)m(ximation)35 b(With)g(Resp)s(ect)g(to)g(the)g(Av)m(erage)471 3875 y Fv(Our)44 b(approac)m(h)h(is)f(to)g(lo)s(ok)f(at)g(appro)m(ximation)f (results)j(with)f(resp)s(ect)h(to)f(the)h(a)m(v)m(erage)330 4092 y(function)g(v)-5 b(alue)46 b(o)m(v)m(er)g(all)e(p)s(erm)m (utations.)82 b(In)47 b(the)f(cases)h(of)e Fk(TSP)i Fv(and)f Fk(QAP)p Fv(,)g(it)f(is)g(easy)i(to)330 4308 y(compute)35 b(the)h(a)m(v)m(erage)h(o)m(v)m(er)f(all)d(p)s(erm)m(utations,)j(see)g (Lemma)e(3.1.1.)52 b(Equiv)-5 b(alen)m(tly)d(,)35 b(w)m(e)i(can)330 4525 y(consider)29 b(a)g(normalized)d(maximization)f(problem)j(with)g (a)m(v)m(erage)i(v)-5 b(alue)28 b(on)h(all)e(p)s(erm)m(utations)330 4742 y(of)32 b(0.)43 b(In)33 b(this)f(case)i(some)e(edge)i(w)m(eigh)m (ts)f(will)d(b)s(e)j(negativ)m(e.)471 4958 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3225 y(Less)38 b(empirical)c(w)m(ork)j (has)h(b)s(een)f(done)g(on)g(non-Euclidean)f(and)g(asymmetric)g (instances)330 3442 y(of)31 b(the)g Fk(TSP)p Fv(.)h(An)g(in)m (teresting)f(surv)m(ey)i(of)e(asymmetric)f Fk(TSP)p Fv('s)i(dra)m(wn)g (from)e(v)-5 b(arious)31 b(sources)h(is)330 3658 y([MP91].)58 b(In)38 b(these)g(exp)s(erimen)m(ts,)i(v)-5 b(arious)36 b(asymmetric)g Fk(TSP)p Fv('s)j(are)e(solv)m(ed,)j(some)d(random,)330 3875 y(some)25 b(structured.)42 b(The)26 b(authors)e(rep)s(ort)h (success)i(in)d(solving)g(randomly)f(generated)j(problems)330 4092 y(on)h(thousands)g(of)g(no)s(des)g(using)f(branc)m(h-and-b)s(ound) h(heuristics,)h(as)f(w)m(ell)f(as)h(some)f(structured)330 4308 y(problems)32 b(of)g(similar)d(sizes.)471 4525 y(More)e(recen)m (tly)h([CJMZ01])f(studied)g(heuristics)g(on)f(a)h(v)-5 b(ariet)m(y)26 b(of)g(di\016cult)g(\(some)h(random,)330 4742 y(some)36 b(structured\))i(asymmetric)d Fk(TSP)i Fv(instances.)55 b(They)38 b(routinely)d(solv)m(e)i(problems)e(on)i (100)330 4958 y(no)s(des,)c(and)g(are)g(usually)e(able)h(to)h(solv)m(e) g(problems)e(of)i(size)f(1000.)p eop %%Page: 15 23 15 22 bop 2070 -210 a Fw(15)330 91 y Fs(1.2.4)105 b(Non-p)s(olynomial) 35 b(Algorithms)471 372 y Fv(T)-8 b(o)42 b(help)f(understand)h(the)g (complexit)m(y)f(of)g(problems,)h(w)m(e)h(can)e(also)g(consider)g(the)h (p)s(er-)330 589 y(formance)33 b(of)g(non-p)s(olynomial)d(algorithms.) 44 b(The)34 b Fk(TSP)h Fv(can)f(b)s(e)g(solv)m(ed)g(in)f(time)f Fl(O)s Fv(\()p Fl(n)3591 553 y Fo(2)3630 589 y Fv(2)3679 553 y Fp(n)3725 589 y Fv(\))i(b)m(y)330 806 y(dynamic)g(programming.)47 b(This)35 b(is)f(not)g(p)s(olynomial,)e(but)j(is)f(substan)m(tially)g (faster)h(than)f(the)330 1022 y Fl(O)s Fv(\()p Fl(n)p Fv(!\))e(required)h(to)f(en)m(umerate)h(the)g(solutions.)330 1289 y Ft(1.3)112 b(Remarks)37 b(on)h(the)f Fk(QAP)471 1604 y Fv(Lik)m(e)31 b(the)h Fk(TSP)p Fv(,)g(the)g Fk(QAP)g Fv(is)f(kno)m(wn)h(as)g(one)f(of)g(the)h(most)e(di\016cult)g(problems)h (in)f(com)m(bina-)330 1821 y(torial)g(optimization.)40 b(A)33 b(go)s(o)s(d)e(recen)m(t)j(surv)m(ey)h(of)d(the)h Fk(QAP)g Fv(literature)e(is)i([BC)-56 b(\030)13 b(PP99].)330 2073 y Fs(1.3.1)105 b(Complexit)m(y)34 b(Results)i(for)f Fq(QAP)471 2354 y Fv(W)-8 b(e)37 b(noted)f(in)f(Section)h(1.2)g(that)g (the)h Fk(QAP)f Fv(generalizes)g(the)h Fk(TSP)p Fv(.)g(Hence)g Fk(QAP)g Fv(is)f(at)f(least)330 2571 y(as)30 b(hard)h(as)f Fk(TSP)p Fv(,)h(and)g(the)g(hardness)g(results)g(w)m(e)g(ha)m(v)m(e)h (for)e Fk(TSP)h Fv(apply)f(to)g Fk(QAP)p Fv(.)h(F)-8 b(or)29 b(example,)330 2788 y(minimizing)f(the)33 b Fk(QAP)g Fv(with)g(p)s(ositiv)m(e)e(w)m(eigh)m(ts)j(is)e(NPO-complete)g([A)m(CG) 3121 2751 y Fo(+)3180 2788 y Fv(99].)471 3004 y(In)f(fact,)h(the)f Fk(QAP)h Fv(app)s(ears)f(m)m(uc)m(h)h(more)e(di\016cult)g(than)h(the)h Fk(TSP)g Fv(b)s(oth)e(theoretically)g(and)330 3221 y(practically)-8 b(.)40 b(There)29 b(are)f(few)g(p)s(ositiv)m(e)g(appro)m(ximation)d (results)k(for)e(sp)s(ecial)g(cases)i(of)f(the)g Fk(QAP)p Fv(.)471 3438 y(F)-8 b(or)45 b(the)g(problem)f(of)h(maximizing)c(a)k Fk(QAP)h Fv(with)f(p)s(ositiv)m(e)g(co)s(e\016cien)m(ts,)k(w)m(e)e(can) e(get)g(a)330 3654 y(trivial)25 b(appro)m(ximation)g(guaran)m(tee)j(of) f(1)p Fl(=n)1964 3618 y Fo(2)2031 3654 y Fv(b)m(y)h(pic)m(king)f(a)g(p) s(erm)m(utation)f(whic)m(h)i(includes)f(the)330 3871 y(largest)k(p)s(ossible)h(single)f(term)g(in)h(the)g(ob)5 b(jectiv)m(e)33 b(function.)43 b(Sp)s(eci\014cally)-8 b(,)31 b(w)m(e)i(tak)m(e)g Fl(i)3544 3835 y Ff(0)3568 3871 y Fl(;)17 b(j)3658 3835 y Ff(0)3681 3871 y Fl(;)g(k)3779 3835 y Ff(0)3802 3871 y Fl(;)g(l)3877 3835 y Ff(0)330 4088 y Fv(so)33 b(that:)791 4404 y Fl(a)842 4419 y Fp(i)866 4400 y Fh(0)889 4419 y Fp(j)922 4400 y Fh(0)948 4404 y Fl(b)989 4419 y Fp(k)1028 4400 y Fh(0)1050 4419 y Fp(l)1072 4400 y Fh(0)1126 4404 y Fv(=)28 b(max)1428 4264 y Fj(\032\022)1576 4404 y Fv(max)1611 4467 y Fp(i)p Ff(6)p Fo(=)p Fp(j)1774 4404 y Fl(a)1825 4419 y Fp(ij)1886 4264 y Fj(\023)17 b(\022)2049 4404 y Fv(max)2082 4467 y Fp(k)r Ff(6)p Fo(=)p Fp(l)2247 4404 y Fl(b)2288 4419 y Fp(k)r(l)2353 4264 y Fj(\023)2443 4404 y Fl(;)2487 4293 y Fj(\020)2546 4404 y Fv(max)2625 4464 y Fp(i)2745 4404 y Fl(a)2796 4419 y Fp(ii)2848 4293 y Fj(\021)g(\020)2984 4404 y Fv(max)3055 4467 y Fp(k)3182 4404 y Fl(b)3223 4419 y Fp(k)r(k)3304 4293 y Fj(\021)3364 4264 y(\033)330 4720 y Fv(W)-8 b(e)41 b(can)g(c)m(ho)s(ose)h Fl(i)1044 4684 y Ff(0)1068 4720 y Fl(;)17 b(j)1158 4684 y Ff(0)1181 4720 y Fl(;)g(k)1279 4684 y Ff(0)1302 4720 y Fl(;)g(l)1377 4684 y Ff(0)1441 4720 y Fv(so)41 b(that)g(either)f Fl(i)2106 4684 y Ff(0)2172 4720 y Fu(6)p Fv(=)h Fl(j)2335 4684 y Ff(0)2399 4720 y Fv(and)g Fl(k)2651 4684 y Ff(0)2717 4720 y Fu(6)p Fv(=)g Fl(l)2865 4684 y Ff(0)2930 4720 y Fv(or)f Fl(i)3090 4684 y Ff(0)3156 4720 y Fv(=)h Fl(j)3319 4684 y Ff(0)3383 4720 y Fv(and)g Fl(k)3635 4684 y Ff(0)3701 4720 y Fv(=)g Fl(l)3849 4684 y Ff(0)3873 4720 y Fv(.)330 4937 y(Then)33 b(for)f(an)m(y)h(p)s(erm)m(utation)e Fl(\033)37 b Fv(suc)m(h)c(that)f Fl(\033)t Fv(\()p Fl(i)2127 4901 y Ff(0)2151 4937 y Fv(\))c(=)f Fl(k)2374 4901 y Ff(0)2398 4937 y Fv(,)32 b(and)g(if)g Fl(i)2769 4901 y Ff(0)2820 4937 y Fu(6)p Fv(=)27 b Fl(j)2969 4901 y Ff(0)2993 4937 y Fv(,)32 b(suc)m(h)i(that)e Fl(\033)t Fv(\()p Fl(j)3626 4901 y Ff(0)3649 4937 y Fv(\))c(=)f Fl(l)3849 4901 y Ff(0)3873 4937 y Fv(,)330 5154 y(w)m(e)35 b(ha)m(v)m(e)h Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))30 b Fu(\025)h Fl(f)11 b Fv(\()p Fl(\034)g Fv(\))p Fl(=n)1329 5118 y Fo(2)1368 5154 y Fv(.)49 b(By)34 b(\014xing)g(more)g(terms,)g (this)g(guaran)m(tee)h(can)g(b)s(e)f(impro)m(v)m(ed)g(b)m(y)330 5370 y(a)39 b(constan)m(t)h(factor)e(\(at)h(the)g(price)g(of)g(a)g(w)m (orse)h(p)s(olynomial)35 b(running)k(time\),)g(but)h(w)m(e)g(kno)m(w)p eop %%Page: 16 24 16 23 bop 2070 -210 a Fw(16)330 91 y Fv(of)36 b(no)g(algorithm)c(in)k (the)g(literature)f(that)h(impro)m(v)m(es)g(on)g(this)g(guaran)m(tee)g (for)g(all)e Fk(QAP)p Fv('s)j(with)330 308 y(p)s(ositiv)m(e)32 b(w)m(eigh)m(ts.)471 525 y(Arkin,)40 b(Hassin)f(and)g(Sviridenk)m(o)f ([AHS00])h(consider)g(the)g(problem)e(of)h(maximizing)d(the)330 741 y Fk(QAP)47 b Fv(where)h(one)f(of)f(the)h(matrices)f(is)g(assumed)h (to)f(satisfy)h(the)g(triangle)e(inequalit)m(y)-8 b(,)49 b(and)330 958 y(the)35 b(other)h(is)e(non-negativ)m(e)h(\(analogous)f (to)h(a)f(metric)g Fk(TSP)p Fv(\).)i(They)h(get)e(a)3204 919 y Fo(1)p 3204 935 36 4 v 3204 992 a(4)3250 958 y Fv(-appro)m(ximation)330 1175 y(guaran)m(tee.)48 b(It)33 b(is)h(NP-complete)f(to)g(get)h(a)f(constan)m(t)i(appro)m(ximation)c (for)j(the)g(minimization)330 1391 y(v)m(ersion)f(of)f(this)g(problem)g ([Que86].)471 1608 y(Y)-8 b(e)22 b([Y)-8 b(e99])22 b(giv)m(es)g(an)g (algorithm)c(using)k(semi-de\014nite)e(programming)f(that)j(giv)m(es)g (a)3757 1541 y(1)p 3572 1585 418 4 v 3572 1676 a Fl(n)3630 1648 y Fo(2)3687 1676 y Fv(ln)o(\(4)p Fl(n)3913 1648 y Fo(4)3952 1676 y Fv(\))330 1825 y(appro)m(ximation)43 b(guaran)m(tee)i(for)g(maximizing)c(\(1.2\),)47 b(when)f(the)g(tensor)f Fl(C)56 b Fv(=)3428 1714 y Fj(n)3494 1825 y Fl(c)3536 1778 y Fp(ij)3536 1853 y(k)r(l)3649 1825 y Fv(:)49 b(1)f Fu(\024)330 2042 y Fl(i;)17 b(j;)g(k)s(;)g(l)30 b Fu(\024)e Fl(n)811 1931 y Fj(o)910 2042 y Fv(is)k(p)s(ositiv)m(e)g (semi-de\014nite)g(when)i(considered)f(as)g(an)f Fl(n)2984 2005 y Fo(2)3046 2042 y Fu(\002)23 b Fl(n)3204 2005 y Fo(2)3276 2042 y Fv(matrix.)471 2258 y(There)i(is)e(another)h(notion)e (of)h(a)g(\\go)s(o)s(d")f(appro)m(ximation)f(called)i(the)h(domination) c(n)m(um)m(b)s(er.)330 2475 y(The)40 b Fm(domination)g(numb)-5 b(er)39 b Fv(of)g(a)h(p)s(erm)m(utation)e Fl(\033)43 b Fv(is)c(the)h(n)m(um)m(b)s(er)g(of)f(p)s(erm)m(utations)g(whose)330 2692 y(ob)5 b(jectiv)m(e)33 b(v)-5 b(alue)31 b(is)g(at)g(most)g Fl(f)11 b Fv(\()p Fl(\033)t Fv(\).)43 b(F)-8 b(or)31 b(example,)g(if)g(the)h(ob)5 b(jectiv)m(e)33 b(function)e(is)g (maximized)330 2908 y(at)g Fl(\034)11 b Fv(,)32 b(then)g Fl(\034)43 b Fv(has)32 b(domination)d(n)m(um)m(b)s(er)i Fl(n)p Fv(!.)44 b(The)32 b(problem)e(of)h(\014nding)g(a)g(p)s(erm)m (utation)f(with)330 3125 y(go)s(o)s(d)41 b(domination)f(n)m(um)m(b)s (er)j(for)e(the)i Fk(QAP)g Fv(is)f(examined)g(in)g([GY02)o(].)73 b(They)44 b(sho)m(w)f(that)f(at)330 3342 y(least)g(in)h(the)g(case)h (where)g Fl(n)f Fv(is)f(a)h(prime)f(p)s(o)m(w)m(er,)k(there)e(is)e(a)h (p)s(olynomial)c(algorithm)h(that)330 3558 y(returns)c(a)g(p)s(erm)m (utation)e(whose)i(domination)d(n)m(um)m(b)s(er)j(is)f(at)g(least)g(\() p Fl(n)24 b Fu(\000)h Fv(2\)!.)52 b(Additionally)-8 b(,)330 3775 y(they)29 b(sho)m(w)h(that)e(suc)m(h)i(an)e(algorithm)d(exists)k (for)f Fk(TSP)h Fv(for)f(all)f Fl(n)p Fv(.)42 b(W)-8 b(e)29 b(discuss)g(their)f(algorithm)330 3992 y(in)k(Section)g(4.4.)330 4244 y Fs(1.3.2)105 b(Empirical)35 b(Results)471 4525 y Fv(The)e(con)m(trast)f(b)s(et)m(w)m(een)h Fk(QAP)f Fv(and)g Fk(TSP)g Fv(in)f(empirical)d(tests)33 b(is)e(ev)m(en)i(sharp)s (er.)44 b(A)31 b(problem)330 4742 y(library)f(has)j(b)s(een)g(dev)m (elop)s(ed)g(for)e Fk(QAP)p Fv(,)i(called)e Fm(QAPLIB)g Fv([BKR97].)43 b(Most)33 b(of)e(the)i(problems)330 4958 y(in)23 b(the)h(QAPLIB)g(ha)m(v)m(e)h(a)f(simple)e(structure,)27 b(sa)m(y)e(that)e(the)h(distance)g(matrix)f(is)g(obtained)g(from)330 5175 y(an)28 b(arrangemen)m(t)f(of)g(p)s(oin)m(ts)g(in)g(the)h(plane,)h (and)e(some)h(comp)s(onen)m(t)f(that)h(is)f(either)g(random)g(or)p eop %%Page: 17 25 17 24 bop 2070 -210 a Fw(17)330 91 y Fv(dra)m(wn)32 b(from)d(applied)h (data.)43 b(Despite)30 b(considerable)h(e\013ort,)g(it)f(has)i(pro)m(v) m(ed)g(di\016cult)e(to)g(solv)m(e)330 308 y(instances)42 b(of)e(size)h Fl(n)i Fv(=)e(20,)i(and)e(solutions)f(of)h(problems)f(of) g(size)i Fl(n)g Fv(=)g(30)e(are)h(considered)330 525 y(remark)-5 b(able.)42 b(See)34 b(for)e(example)g([ABGL02)o(])h(and)f ([BMCP98)q(].)471 741 y(The)41 b(heuristics)f(that)f(are)h(e\013ectiv)m (e)h(in)e(solving)f(mo)s(derately)h(sized)h Fk(TSP)p Fv('s)h(do)e(not)h(w)m(ork)330 958 y(as)f(w)m(ell)f(on)g Fk(QAP)p Fv('s.)63 b(One)39 b(of)f(the)h(goals)f(of)g(this)g(thesis)h (is)f(to)h(pro)m(vide)f(some)h(explanation)e(of)330 1175 y(wh)m(y)29 b(this)e(is,)h(and)g(to)f(distinguish)f(the)i(cases)g (where)h(w)m(e)g(migh)m(t)d(exp)s(ect)i(heuristics)g(to)f(p)s(erform) 330 1391 y(w)m(ell.)471 1608 y(In)e([AZ01)o(],)i(the)e(authors)g (compare)f(the)h Fk(QAP)g Fv(and)g(the)g Fk(TSP)g Fv(via)f(a)g (\\ruggedness)i(co)s(e\016cien)m(t")330 1825 y(whic)m(h)37 b(describ)s(es)h(a)f(t)m(yp)s(e)h(of)e(lo)s(cal)f(v)-5 b(ariabilit)m(y)d(.)53 b(They)38 b(tak)m(e)g(the)f(view)g(that)g (problems)f(with)330 2042 y(lo)m(w)m(er)f(lo)s(cal)d(v)-5 b(ariabilit)m(y)31 b(are)k(easier)f(for)g(lo)s(cal)e(searc)m(h.)51 b(They)35 b(sho)m(w)h(that)e(the)h(ruggedness)h(of)330 2258 y(the)28 b Fk(QAP)g Fv(lies)f(b)s(et)m(w)m(een)i(the)f(ruggedness) i(of)d(the)h Fk(TAP)g Fv(\(whic)m(h)g(they)g(consider)g(an)g(easy)g (problem)330 2475 y(for)36 b(lo)s(cal)f(searc)m(h\))j(and)g(the)f (ruggedness)i(of)d(the)i(binary)e(string)h(problem,)g(whic)m(h)g(they)h 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Fe(i)2187 4733 y Fo(\))2231 4612 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))28 b Fu(\025)g Fl(\013)330 4937 y Fv(for)k(some)g(n)m (um)m(b)s(er)h Fl(\013)q Fv(,)g(then)1789 5086 y(1)p 1742 5131 145 4 v 1742 5222 a Fu(j)p Fl(X)8 b Fu(j)1917 5059 y Fj(X)1912 5270 y Fp(\033)r Ff(2)p Fp(X)2082 5154 y Fl(f)j Fv(\()p Fl(\033)t Fv(\))27 b Fu(\025)h Fl(\013)q(:)p eop %%Page: 37 45 37 44 bop 2070 -210 a Fw(37)330 91 y Ft(De\014nition)36 b(3.2.2.)49 b Fv(A)29 b(\(linear\))e Fm(r)-5 b(epr)g(esentation)28 b Fv(of)g(a)h(group)f Fl(G)h Fv(is)f(a)g(group)h(homomorphism)330 308 y Fl(\032)41 b Fv(:)f Fl(G)h Fu(!)f Fl(GL)p Fv(\()p Fl(V)22 b Fv(\))39 b(where)j Fl(GL)p Fv(\()p Fl(V)22 b Fv(\))40 b(is)f(the)i(group)f(of)f(in)m(v)m(ertible)g(linear)g (transformations)f(of)330 525 y(a)f(suitable)f(v)m(ector)i(space)h Fl(V)21 b Fv(.)58 b(The)38 b Fm(de)-5 b(gr)g(e)g(e)36 b Fv(of)h(the)g(represen)m(tation)h(is)f(the)h(dimension)e(of)g Fl(V)22 b Fv(.)330 741 y(Equiv)-5 b(alen)m(tly)d(,)31 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b(the)f(submo)s(dule)e(that)i(is)e(the)i(sum)g(of)e(all)g (the)i(\(isomorphic\))d(submo)s(dules)330 3850 y(asso)s(ciated)25 b(to)g(a)g(giv)m(en)h(irreducible)e(represen)m(tation)i(is)f(unique.)42 b(This)25 b(is)g(called)g(the)g Fm(isotypic)-5 b(al)330 4066 y(c)g(omp)g(onent)31 b Fv(of)h(the)h(represen)m(tation.)45 b(W)-8 b(e)33 b(refer)g(to)f([FH91)o(])h(for)f(further)h(details.)471 4283 y(W)-8 b(e)33 b(aim)d(to)i(understand)i(the)e(conjugation)f (represen)m(tation)i(b)m(y)g(decomp)s(osing)f(it)f(in)m(to)g(its)330 4500 y(isot)m(ypical)i(comp)s(onen)m(ts.)50 b(In)35 b(particular,)f (this)g(decomp)s(osition)f(will)g(allo)m(w)g(us)i(to)f(determine)330 4716 y(the)f(cen)m(tral)f(pro)5 b(jection.)471 4933 y(W)-8 b(e)43 b(describ)s(e)f(the)h(in)m(v)-5 b(arian)m(t)41 b(subspaces)j(of)e(the)g(action)f(of)h Fl(S)2868 4948 y Fp(n)2957 4933 y Fv(in)f(the)i(space)g(of)f Fl(n)29 b Fu(\002)g Fl(n)330 5150 y Fv(matrices)46 b(Mat)911 5165 y Fp(n)1005 5150 y Fv(b)m(y)i(sim)m(ultaneous)e(p)s(erm)m (utations)g(of)h(ro)m(ws)h(and)f(columns.)86 b(F)-8 b(or)46 b Fl(n)52 b Fu(\025)h Fv(4,)330 5367 y(there)29 b(are)f(sev)m(en)j(suc) m(h)e(in)m(v)-5 b(arian)m(t)27 b(subspaces,)32 b(and)c(w)m(e)h(ha)m(v)m (e)h(group)s(ed)e(isomorphic)e(subspaces)p eop %%Page: 38 46 38 45 bop 2070 -210 a Fw(38)330 91 y Fv(together)34 b(to)f(giv)m(e)g (the)h(four)f(isot)m(ypical)f(comp)s(onen)m(ts)i(of)e(the)i(represen)m (tation.)47 b(Our)33 b(notation)330 308 y(is)g(inspired)g(b)m(y)i(the)f (generally)e(accepted)j(notation)d(of)h(represen)m(tation)i(theory)f (of)f Fl(S)3543 323 y Fp(n)3590 308 y Fv(,)h(where)330 525 y(irreducible)e(represen)m(tations)j(and)e(their)g(asso)s(ciated)h (subspaces)h(are)f(indexed)g(b)m(y)h(partitions)330 741 y(of)d(the)h(in)m(teger)g Fl(n)p Fv(.)330 994 y Fs(Subspace)j Fr(L)846 1008 y Fp(n)471 1275 y Fv(Let)d Fl(L)712 1239 y Fo(1)712 1299 y Fp(n)792 1275 y Fv(b)s(e)g(the)g(space)g(of)f (constan)m(t)i(matrices)d Fl(A)p Fv(:)1148 1591 y Fl(a)1199 1606 y Fp(ij)1287 1591 y Fv(=)d Fl(\013)98 b Fv(for)32 b(some)97 b Fl(\013)h Fv(and)33 b(all)96 b(1)27 b Fu(\024)h Fl(i;)17 b(j)34 b Fu(\024)28 b Fl(n:)330 1907 y Fv(Let)33 b Fl(L)571 1871 y Fo(2)571 1932 y Fp(n)651 1907 y Fv(b)s(e)f(the)h (subspace)i(of)d(scalar)g(matrices)g Fl(A)p Fv(:)1367 2334 y Fl(a)1418 2349 y Fp(ij)1506 2334 y Fv(=)1610 2040 y Fj(8)1610 2130 y(>)1610 2159 y(>)1610 2189 y(>)1610 2219 y(<)1610 2399 y(>)1610 2428 y(>)1610 2458 y(>)1610 2488 y(:)1698 2231 y Fl(\013)98 b Fv(if)64 b Fl(i)28 b Fv(=)g Fl(j)1698 2491 y Fv(0)111 b(if)64 b Fl(i)28 b Fu(6)p Fv(=)g Fl(j)2315 2334 y Fv(for)k(some)97 b Fl(\013)q(:)330 2765 y Fv(Finally)-8 b(,)38 b(Let)i Fl(L)937 2780 y Fp(n)1024 2765 y Fv(=)f Fl(L)1205 2729 y Fo(1)1205 2789 y Fp(n)1279 2765 y Fv(+)27 b Fl(L)1448 2729 y Fo(2)1448 2789 y Fp(n)1495 2765 y Fv(.)64 b(One)40 b(can)f(observ)m(e)j(that)d(dim)15 b Fl(L)2802 2780 y Fp(n)2889 2765 y Fv(=)39 b(2)g(and)h(that)f Fl(L)3573 2780 y Fp(n)3660 2765 y Fv(is)f(the)330 2981 y(subspace)d(of)d(all)e(matrices)i(that)g(remain)f(\014xed)j(under)g (the)f(action)e(of)h Fl(S)3055 2996 y Fp(n)3102 2981 y Fv(.)330 3234 y Fs(Subspace)k Fr(L)846 3248 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)471 3515 y Fv(Let)29 b Fl(L)708 3479 y Fo(1)708 3539 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)930 3515 y Fv(b)s(e)g(the)g(subspace)i(of)e(matrices)f(with)g (iden)m(tical)g(ro)m(ws)h(and)g(suc)m(h)i(that)e(the)g(sum)330 3732 y(of)j(en)m(tries)h(in)f(eac)m(h)i(ro)m(w)f(is)f(0:)909 4298 y Fl(A)27 b Fv(=)1113 3858 y Fj(0)1113 4034 y(B)1113 4093 y(B)1113 4153 y(B)1113 4213 y(B)1113 4273 y(B)1113 4333 y(B)1113 4392 y(B)1113 4452 y(B)1113 4516 y(@)1215 3991 y Fl(\013)1277 4006 y Fo(1)1429 3991 y Fl(\013)1491 4006 y Fo(2)1628 3991 y Fl(:)17 b(:)g(:)102 b(\013)1907 4006 y Fp(n)1215 4208 y Fl(\013)1277 4223 y Fo(1)1429 4208 y Fl(\013)1491 4223 y Fo(2)1628 4208 y Fl(:)17 b(:)g(:)102 b(\013)1907 4223 y Fp(n)1200 4425 y Fl(:)17 b(:)g(:)99 b(:)17 b(:)g(:)99 b(:)17 b(:)g(:)100 b(:)17 b(:)g(:)1215 4641 y(\013)1277 4656 y Fo(1)1429 4641 y Fl(\013)1491 4656 y Fo(2)1628 4641 y Fl(:)g(:)g(:)102 b(\013)1907 4656 y Fp(n)1957 3858 y Fj(1)1957 4034 y(C)1957 4093 y(C)1957 4153 y(C)1957 4213 y(C)1957 4273 y(C)1957 4333 y(C)1957 4392 y(C)1957 4452 y(C)1957 4516 y(A)2061 4298 y Fl(;)114 b Fv(where)99 b Fl(\013)2611 4313 y Fo(1)2673 4298 y Fv(+)22 b Fl(:)17 b(:)g(:)k Fv(+)h Fl(\013)3067 4313 y Fp(n)3142 4298 y Fv(=)27 b(0)p Fl(:)p eop %%Page: 39 47 39 46 bop 2070 -210 a Fw(39)330 91 y Fv(Similarly)-8 b(,)41 b(let)h Fl(L)993 55 y Fo(2)993 116 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1228 91 y Fv(b)s(e)h(the)g(subspace)h(of)e(matrices) g(with)g(iden)m(tical)f(columns)h(and)g(suc)m(h)330 308 y(that)32 b(the)h(sum)g(of)f(en)m(tries)h(in)f(eac)m(h)h(column)f(is)g (0:)909 874 y Fl(A)27 b Fv(=)1113 435 y Fj(0)1113 610 y(B)1113 670 y(B)1113 730 y(B)1113 790 y(B)1113 849 y(B)1113 909 y(B)1113 969 y(B)1113 1029 y(B)1113 1092 y(@)1215 568 y Fl(\013)1277 583 y Fo(1)1429 568 y Fl(\013)1491 583 y Fo(1)1628 568 y Fl(:)17 b(:)g(:)106 b(\013)1911 583 y Fo(1)1215 784 y Fl(\013)1277 799 y Fo(2)1429 784 y Fl(\013)1491 799 y Fo(2)1628 784 y Fl(:)17 b(:)g(:)106 b(\013)1911 799 y Fo(2)1200 1001 y Fl(:)17 b(:)g(:)99 b(:)17 b(:)g(:)99 b(:)17 b(:)g(:)100 b(:)17 b(:)g(:)1211 1218 y(\013)1273 1233 y Fp(n)1425 1218 y Fl(\013)1487 1233 y Fp(n)1628 1218 y Fl(:)g(:)g(:)102 b(\013)1907 1233 y Fp(n)1957 435 y Fj(1)1957 610 y(C)1957 670 y(C)1957 730 y(C)1957 790 y(C)1957 849 y(C)1957 909 y(C)1957 969 y(C)1957 1029 y(C)1957 1092 y(A)2061 874 y Fl(;)114 b Fv(where)99 b Fl(\013)2611 889 y Fo(1)2673 874 y Fv(+)22 b Fl(:)17 b(:)g(:)k Fv(+)h Fl(\013)3067 889 y Fp(n)3142 874 y Fv(=)27 b(0)p Fl(:)330 1470 y Fv(Finally)-8 b(,)32 b(let)i Fl(L)892 1433 y Fo(3)892 1494 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1119 1470 y Fv(b)s(e)g(the)h(subspace)i(of)d (diagonal)e(matrices)h(whose)j(diagonal)c(en)m(tries)j(sum)330 1686 y(to)d(zero:)802 2252 y Fl(A)27 b Fv(=)1006 1813 y Fj(0)1006 1988 y(B)1006 2048 y(B)1006 2108 y(B)1006 2168 y(B)1006 2228 y(B)1006 2287 y(B)1006 2347 y(B)1006 2407 y(B)1006 2471 y(@)1108 1946 y Fl(\013)1170 1961 y Fo(1)1348 1946 y Fv(0)124 b Fl(:)17 b(:)g(:)141 b Fv(0)156 b(0)1134 2163 y(0)139 b Fl(\013)1384 2178 y Fo(2)1521 2163 y Fl(:)17 b(:)g(:)141 b Fv(0)156 b(0)1093 2379 y Fl(:)17 b(:)g(:)99 b(:)17 b(:)g(:)99 b(:)17 b(:)g(:)99 b(:)17 b(:)g(:)100 b(:)17 b(:)g(:)1134 2596 y Fv(0)165 b(0)124 b Fl(:)17 b(:)g(:)141 b Fv(0)126 b Fl(\013)2014 2611 y Fp(n)2064 1813 y Fj(1)2064 1988 y(C)2064 2048 y(C)2064 2108 y(C)2064 2168 y(C)2064 2228 y(C)2064 2287 y(C)2064 2347 y(C)2064 2407 y(C)2064 2471 y(A)2168 2252 y Fl(;)114 b Fv(where)99 b Fl(\013)2718 2267 y Fo(1)2780 2252 y Fv(+)22 b Fl(:)17 b(:)g(:)k Fv(+)h Fl(\013)3174 2267 y Fp(n)3249 2252 y Fv(=)27 b(0)p Fl(:)330 2848 y Fv(Let)41 b Fl(L)579 2863 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)812 2848 y Fv(=)g Fl(L)995 2812 y Fo(1)995 2872 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1215 2848 y Fv(+)27 b Fl(L)1384 2812 y Fo(2)1384 2872 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1604 2848 y Fv(+)g Fl(L)1773 2812 y Fo(3)1773 2872 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1965 2848 y Fv(.)67 b(One)41 b(can)g(c)m(hec)m(k)h(that)e(the)h(dimension)e (of)h(eac)m(h)330 3065 y(of)d Fl(L)512 3028 y Fo(1)512 3089 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)705 3065 y Fv(,)i Fl(L)837 3028 y Fo(2)837 3089 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1067 3065 y Fv(and)f Fl(L)1328 3028 y Fo(3)1328 3089 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1558 3065 y Fv(is)f Fl(n)26 b Fu(\000)g Fv(1)38 b(and)f(that)h(dim)15 b Fl(L)2591 3080 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2820 3065 y Fv(=)36 b(3)p Fl(n)26 b Fu(\000)g Fv(3.)59 b(Moreo)m(v)m(er,)40 b(the)330 3281 y(subspaces)g Fl(L)847 3245 y Fo(1)847 3306 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1039 3281 y Fv(,)f Fl(L)1171 3245 y Fo(2)1171 3306 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1401 3281 y Fv(and)f Fl(L)1662 3245 y Fo(3)1662 3306 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1892 3281 y Fv(do)g(not)f(con)m(tain)g(non-trivial)e(in)m(v)-5 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Ft(De\014nition)36 b(3.2.3.)49 b Fv(The)40 b Fm(char)-5 b(acter)39 b Fv(of)f(a)h(represen)m(tation)h Fl(\032)f Fv(is)g(the)g(function)g Fl(\037)f Fv(:)h Fl(G)g Fu(!)f Fi(C)p eop %%Page: 41 49 41 48 bop 2070 -210 a Fw(41)330 91 y Fv(giv)m(en)33 b(b)m(y)g Fl(\037)p Fv(\()p Fl(\033)t Fv(\))28 b(=)f(trace)q(\()p Fl(\032)p Fv(\()p Fl(\033)t Fv(\)\).)471 353 y(The)34 b(c)m(haracter)f(is)f(w)m(ell-de\014ned,)h(since)g(trace)g(is)f(in)m(v) -5 b(arian)m(t)31 b(under)j(c)m(hange)f(of)f(basis.)471 570 y(Of)38 b(particular)e(in)m(terest)i(are)g(the)g(c)m(haracters)h (of)e(the)h(irreducible)f(represen)m(tations)i(asso-)330 786 y(ciated)i(to)g(the)h(isot)m(ypical)e(subspaces)k(describ)s(ed)e (ab)s(o)m(v)m(e.)71 b(It)42 b(turns)g(out)f(that)h(w)m(e)g(can)g(giv)m (e)330 1003 y(form)m(ulas)36 b(for)h(these)h(irreducible)e(c)m (haracters)j(in)d(terms)h(of)g(the)h(functions)f Fl(p)p Fv(\()p Fl(\033)t Fv(\))g(\(n)m(um)m(b)s(er)g(of)330 1220 y(\014xed)29 b(p)s(oin)m(ts\))e(and)h Fl(t)p Fv(\()p Fl(\033)t 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b Fv(is)330 4497 y(iden)m(ti\014ed)37 b(with)f(the)i(\(real\))e(group)g(algebra)g(of)h(the)g(symmetric)f (group.)57 b(The)38 b(cen)m(ter)g(of)e(the)330 4714 y(group)e(algebra)g (is)g(spanned)i(b)m(y)f(the)g(c)m(haracters)h(of)f(the)g(irreducible)e (represen)m(tations)j(of)e Fl(S)3826 4729 y Fp(n)3873 4714 y Fv(.)330 4931 y(The)28 b(basic)f(fact)g(that)g(w)m(e)h(are)f (using)g(here)h(is)f(that)g(if)f Fl(f)38 b Fv(is)26 b(a)h(matrix)f (elemen)m(t)h(in)f(an)h(irreducible)330 5147 y(represen)m(tation)35 b(of)e(the)h(group)g(then)g(the)h(cen)m(tral)e(pro)5 b(jection)34 b(\(De\014nition)e(3.1.3\))h(m)m(ust)h(b)s(e)g(a)330 5364 y(scalar)e(m)m(ultiple)e(of)i(the)h(c)m(haracter)h(of)e(that)g (represen)m(tation.)p eop %%Page: 43 51 43 50 bop 2070 -210 a Fw(43)330 91 y Fm(R)-5 b(emark)34 b(3.2.5.)48 b Fv(W)-8 b(e)31 b(note)f(that)g(the)g(functions)g Fl(p;)17 b(p)2293 55 y Fo(2)2362 91 y Fv(and)30 b Fl(t)g Fv(are)g(ob)5 b(jectiv)m(e)31 b(functions)f(of)g(t)m(yp)s(e)330 308 y(\(1.2\))22 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b(will)d(w)m(ork)j (with)f(the)h(generalized)f(function)g Fl(f)44 b Fv(of)33 b(t)m(yp)s(e)h(\(1.2\),)f(for)g(some)g(tensor)h Fl(C)i Fv(=)330 3641 y(\()p Fl(c)410 3594 y Fp(ij)410 3669 y(k)r(l)474 3641 y Fv(\).)43 b(T)-8 b(o)31 b(compute)g(a)f(particular)f Fl(\015)1696 3656 y Fp(p)1766 3641 y Fv(where)j Fl(p)e Fv(is)h(a)f(partition)f(of)h Fl(n)h Fv(\(in)e(this)i(case)g (necessarily)330 3858 y(one)36 b(of)f Fu(f)p Fl(n)p Fu(g)p Fl(;)17 b Fu(f)p Fl(n)24 b Fu(\000)h Fv(1)p Fl(;)17 b Fv(1)p Fu(g)p Fl(;)g Fu(f)p Fl(n)23 b Fu(\000)i Fv(2)p Fl(;)17 b Fv(2)p Fu(g)35 b Fv(or)g Fu(f)p Fl(n)25 b Fu(\000)g Fv(2)p Fl(;)17 b Fv(1)p Fl(;)g Fv(1)p Fu(g)p Fv(\),)35 b(w)m(e)i(pro)5 b(ject)36 b(the)g(tensor)h Fl(C)42 b Fv(in)m(to)35 b(the)330 4074 y(isot)m(ypical)c(comp)s(onen)m(t)i(asso)s (ciated)g(with)f Fl(p)p Fv(,)h(and)g(then)g(compute)g(the)g(cen)m(tral) g(pro)5 b(jection)32 b(of)330 4291 y(the)39 b(part)g(of)f Fl(f)49 b Fv(attributable)38 b(to)g(this)g(comp)s(onen)m(t.)62 b(F)-8 b(or)38 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b(around)g(the)g(maxim)m(um,)g(whic)m(h)g(ma)m(y)g(not)g(b)s (e)g(lo)s(cated)f(at)g(the)330 5367 y(iden)m(tit)m(y)-8 b(,)34 b(but)h(instead)f(at)f(a)h(p)s(erm)m(utation)f Fl(\034)45 b Fv(that)34 b(w)m(e)h(do)f(not)g(kno)m(w)h(in)f(adv)-5 b(ance.)48 b(Giv)m(en)34 b Fl(\034)11 b Fv(,)p eop %%Page: 45 53 45 52 bop 2070 -210 a Fw(45)330 91 y Fv(w)m(e)37 b(could)e(replace)g Fl(f)46 b Fv(b)m(y)37 b(the)f(shifted)f(function)g Fl(f)2241 55 y Fp(\034)2319 91 y Fv(from)g(Remark)g(3.1.2,)h(whose)g(maxim)m(um) 330 308 y(lies)30 b(at)h(the)g(iden)m(tit)m(y)-8 b(.)43 b(This)31 b(shift)g(is)f(accomplished)g(b)m(y)i(using)f Fl(\034)11 b Fv(\()p Fl(C)c Fv(\))31 b(in)g(place)f(of)h Fl(C)7 b Fv(.)43 b(W)-8 b(e)31 b(then)330 525 y(compute)39 b(the)g(pro)5 b(jection)38 b Fl(P)1438 540 y Fp(p)1478 525 y Fv(\()p Fl(\034)11 b Fv(\()p Fl(C)c Fv(\)\))39 b(and)f(test)i(if)d(the)i(co)s(e\016cien)m(t)h Fl(\015)2978 489 y Fp(\034)2973 549 y(p)3059 525 y Fv(of)f Fl(\037)3238 540 y Fp(p)3316 525 y Fv(in)f(the)h(cen)m(tral)330 741 y(pro)5 b(jection)32 b(of)g Fl(f)963 705 y Fp(\034)952 766 y(p)1039 741 y Fv(is)g(zero.)471 958 y(Since)f(there)f(ma)m(y)h(b)s (e)f(no)g(e\016cien)m(t)h(w)m(a)m(y)h(of)d(\014nding)h Fl(\034)11 b Fv(,)32 b(w)m(e)f(will)d(ha)m(v)m(e)j(to)f(settle)h(for)e (\014nding)330 1175 y(conditions)46 b(suc)m(h)j(that)d Fl(\015)1327 1139 y Fp(\034)1322 1199 y(p)1423 1175 y Fv(=)52 b(0)47 b(for)f(all)f Fl(\034)64 b Fu(2)53 b Fl(S)2245 1190 y Fp(n)2292 1175 y Fv(.)87 b(W)-8 b(e)47 b(use)h(this)f(to)g (deriv)m(e)g(the)h(relaxed)330 1391 y(conditions)32 b(for)g(Theorems)h (2.1.1,)f(2.2.1)g(and)h(2.3.1.)471 1608 y(W)-8 b(e)33 b(b)s(egin)f(b)m(y)h(noting)f(that)g(the)h(function)f Fl(f)2161 1572 y Fp(\034)2150 1633 y(p)2237 1608 y Fv(from)f Fl(P)2530 1623 y Fp(p)2570 1608 y Fv(\()p Fl(\034)11 b Fv(\()p Fl(C)c Fv(\)\))32 b(is)g(giv)m(en)h(b)m(y:)778 1878 y Fl(f)837 1837 y Fp(\034)826 1902 y(p)879 1878 y Fv(\()p Fl(\033)t Fv(\))28 b(:=)1234 1753 y Fp(n)1184 1783 y Fj(X)1173 1993 y Fp(i;j)t Fo(=1)1339 1878 y Fv(\()p Fl(P)1440 1893 y Fp(p)1479 1878 y Fv(\()p Fl(\034)11 b Fv(\()p Fl(C)c Fv(\)\)\))1799 1831 y Fp(ij)1799 1909 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))2083 1878 y Fv(=)2248 1753 y Fp(n)2198 1783 y Fj(X)2186 1993 y Fp(i;j)t Fo(=1)2353 1878 y Fv(\()p Fl(P)2454 1893 y Fp(p)2493 1878 y Fv(\()p Fl(C)g Fv(\)\))2684 1831 y Fp(ij)2684 1909 y(\033)r(\034)h Fo(\()p Fp(i)p Fo(\))p Fp(\033)r(\034)g Fo(\()p Fp(j)t Fo(\)\))3073 1878 y Fv(=)27 b Fl(f)3224 1893 y Fp(p)3264 1878 y Fv(\()p Fl(\033)t(\034)11 b Fv(\))330 2147 y(F)-8 b(rom)36 b(\(3.2\),)h(w)m(e)i(see)f(that)f(the) g(co)s(e\016cien)m(t)h Fl(\015)2071 2111 y Fp(\034)2066 2172 y(p)2151 2147 y Fv(is)f(zero)g(when)h(the)g(cen)m(tral)f(pro)5 b(jection)37 b Fl(g)3742 2111 y Fp(\034)3738 2172 y(p)3821 2147 y Fv(of)330 2364 y Fl(f)389 2328 y Fp(\034)378 2389 y(p)464 2364 y Fv(is)32 b(zero)h(at)g(the)g(iden)m(tit)m(y)f Fl(")p Fv(.)43 b(Observ)m(e)35 b(that:)1656 2634 y Fl(g)1707 2592 y Fp(\034)1703 2658 y(p)1749 2634 y Fv(\()p Fl(")p Fv(\))28 b(=)f Fl(f)2061 2592 y Fp(\034)2050 2658 y(p)2104 2634 y Fv(\()p Fl(")p Fv(\))g(=)h Fl(f)2405 2649 y Fp(p)2445 2634 y Fv(\()p Fl(\034)11 b Fv(\))330 2903 y(So)32 b(w)m(e)i(will)c(lo) s(ok)i(for)g(conditions)f(so)i(that)f Fl(f)1999 2918 y Fp(p)2039 2903 y Fv(\()p Fl(\034)11 b Fv(\))28 b(=)g(0)k(for)g(all)f Fl(\034)39 b Fu(2)28 b Fl(S)2901 2918 y Fp(n)2948 2903 y Fv(.)471 3120 y(In)42 b(the)h(case)g(of)e(the)i(trivial)c(c)m (haracter,)46 b Fl(\037)2119 3135 y Fp(n)2166 3120 y Fv(,)e(this)e(is)f(an)h(easy)h(exercise.)73 b(The)43 b(function)330 3337 y Fl(f)378 3352 y Fp(n)461 3337 y Fv(giv)m(en)37 b(b)m(y)g(the)f(tensor)h Fl(P)1390 3352 y Fp(n)1437 3337 y Fv(\()p Fl(C)7 b Fv(\))36 b(is)g(constan)m(t)h(on)f (all)e(p)s(erm)m(utations)i(with)f(v)-5 b(alue)p 3485 3255 59 4 v 36 w Fl(f)11 b Fv(.)54 b(So)36 b(the)330 3553 y(condition)27 b(w)m(e)i(arriv)m(e)f(at)g(is)p 1374 3472 V 28 w Fl(f)38 b Fv(=)28 b(0.)42 b(F)-8 b(or)27 b(the)i(remaining)d(c)m(haracters,)k(things)e(are)g(not)g(quite)h(as) 330 3770 y(simple.)42 b(In)32 b(the)g(next)g(t)m(w)m(o)h(sections,)f(w) m(e)h(\014nd)f(conditions)f(so)h(that)f(the)h(functions)g Fl(f)3519 3785 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)3743 3770 y Fv(and)330 3987 y Fl(f)378 4002 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)665 3987 y Fv(are)40 b(zero.)65 b(These)42 b(t)m(w)m(o)e(conditions)f(determine)h(the)g(sp)s (ecial)f(cases)i(of)e(Sections)h(2.2)330 4203 y(and)33 b(2.3.)330 4449 y Fs(3.2.3)105 b(Conditions)36 b(Determining)e(the)g (Pure)i(Sp)s(ecial)f(Case)471 4730 y Fv(Consider)h(no)m(w)f(the)h(c)m (haracter)g Fl(\037)1746 4745 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1938 4730 y Fv(.)51 b(W)-8 b(e)35 b(b)s(egin)g(b)m(y)h (calculating)d Fl(P)3149 4745 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)3340 4730 y Fv(\()p Fl(C)7 b Fv(\),)36 b(the)g(pro-)330 4946 y(jection)c(of)g Fl(C)40 b Fv(in)m(to)31 b Fl(L)1131 4961 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1346 4946 y Fu(\012)23 b Fv(Mat)1622 4961 y Fp(n)1669 4946 y Fv(.)471 5163 y(T)-8 b(o)44 b(do)h(this,)i(w)m(e)e(break)g Fl(L)1507 5178 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1744 5163 y Fv(in)m(to)e(its)h(constituen)m(t)h(subspaces,)50 b Fl(L)3174 5127 y Fo(1)3174 5188 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)3366 5163 y Fv(,)d Fl(L)3506 5127 y Fo(2)3506 5188 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)3743 5163 y Fv(and)330 5380 y Fl(L)396 5344 y Fo(3)396 5404 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)588 5380 y Fv(.)c(An)m(y)32 b Fl(W)41 b Fv(=)28 b(\()p Fl(w)1210 5395 y Fp(ij)1270 5380 y Fv(\))g Fu(2)g Fl(L)1496 5395 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1719 5380 y Fv(is)i(the)h(sum)g(of)f(a)h(matrix)e Fl(X)39 b Fv(with)30 b(iden)m(tical)f(ro)m(ws)j(whose)p eop %%Page: 46 54 46 53 bop 2070 -210 a Fw(46)330 91 y Fv(en)m(tries)38 b(sum)f(to)g(zero,)i(a)f(matrix)d Fl(Y)59 b Fv(with)37 b(iden)m(tical)f(columns)g(whose)j(en)m(tries)f(sum)f(to)g(zero,)330 308 y(and)c(a)f(diagonal)f(matrix)g Fl(Z)39 b Fv(with)33 b(the)g(diagonal)d(summing)h(to)i(zero.)44 b(If)33 b(the)g(diagonals)e (of)h Fl(X)8 b Fv(,)330 525 y Fl(Y)57 b Fv(and)35 b Fl(Z)43 b Fv(are)35 b(giv)m(en)h(b)m(y)h(v)m(ectors)g Fl(x)c Fv(=)f(\()p Fl(x)1931 540 y Fp(i)1960 525 y Fv(\),)k Fl(y)g Fv(=)d(\()p Fl(y)2340 540 y Fp(i)2367 525 y Fv(\))j(and)f Fl(z)j Fv(=)33 b(\()p Fl(z)2908 540 y Fp(i)2936 525 y Fv(\))i(resp)s(ectiv)m(ely)-8 b(,)38 b(then)e(w)m(e)330 741 y(can)d(write:)926 1308 y Fl(W)41 b Fv(=)1163 868 y Fj(0)1163 1044 y(B)1163 1103 y(B)1163 1163 y(B)1163 1223 y(B)1163 1283 y(B)1163 1342 y(B)1163 1402 y(B)1163 1462 y(B)1163 1526 y(@)1250 1001 y Fl(x)1305 1016 y Fo(1)1367 1001 y Fv(+)22 b Fl(y)1513 1016 y Fo(1)1574 1001 y Fv(+)g Fl(z)1717 1016 y Fo(1)1942 1001 y Fl(x)1997 1016 y Fo(2)2059 1001 y Fv(+)g Fl(y)2205 1016 y Fo(1)2430 1001 y Fl(:)17 b(:)g(:)209 b(x)2809 1016 y Fp(n)2878 1001 y Fv(+)22 b Fl(y)3024 1016 y Fo(1)1352 1218 y Fl(x)1407 1233 y Fo(1)1469 1218 y Fv(+)g Fl(y)1615 1233 y Fo(2)1840 1218 y Fl(x)1895 1233 y 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3121 y Fp(i)1566 2982 y(n)1516 3012 y Fj(X)1527 3222 y Fp(j)t Fo(=1)1676 3106 y Fl(w)1746 3121 y Fp(ij)1834 3106 y Fv(=)28 b Fl(nx)2051 3121 y Fp(i)2102 3106 y Fv(+)22 b Fl(z)2245 3121 y Fp(i)2501 3106 y Fv(and)2953 2982 y Fp(n)2902 3012 y Fj(X)2913 3222 y Fp(j)t Fo(=1)3063 3106 y Fl(w)3133 3121 y Fp(j)t(i)3221 3106 y Fv(=)27 b Fl(ny)3430 3121 y Fp(i)3481 3106 y Fv(+)22 b Fl(z)3624 3121 y Fp(i)330 3423 y Fv(Then)34 b(w)m(e)f(get:)1067 3739 y Fl(x)1122 3754 y Fp(i)1178 3739 y Fv(=)1448 3672 y(1)p 1291 3716 363 4 v 1291 3807 a Fl(n)p Fv(\()p Fl(n)23 b Fu(\000)f Fv(2\))1681 3569 y Fj( )1760 3739 y Fv(\()p Fl(n)g Fu(\000)h Fv(1\))2131 3615 y Fp(n)2081 3644 y Fj(X)2092 3854 y Fp(j)t Fo(=1)2241 3739 y Fl(w)2311 3754 y Fp(ij)2394 3739 y Fv(+)2542 3615 y Fp(n)2492 3644 y Fj(X)2503 3854 y Fp(j)t Fo(=1)2652 3739 y Fl(w)2722 3754 y Fp(j)t(i)2805 3739 y Fu(\000)f Fl(nw)3032 3754 y Fp(ii)3085 3569 y Fj(!)1070 4160 y Fl(y)1118 4175 y Fp(i)1174 4160 y Fv(=)1444 4092 y(1)p 1287 4137 V 1287 4228 a Fl(n)p Fv(\()p Fl(n)h Fu(\000)g Fv(2\))1677 3989 y Fj( )1756 4160 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\))2127 4035 y Fp(n)2077 4065 y Fj(X)2088 4275 y Fp(j)t Fo(=1)2237 4160 y Fl(w)2307 4175 y Fp(j)t(i)2390 4160 y Fv(+)2538 4035 y Fp(n)2488 4065 y Fj(X)2499 4275 y Fp(j)t Fo(=1)2648 4160 y Fl(w)2718 4175 y Fp(ij)2801 4160 y Fu(\000)f Fl(nw)3028 4175 y Fp(ii)3081 3989 y Fj(!)1261 4534 y Fl(z)1306 4549 y Fp(i)1362 4534 y Fv(=)1603 4467 y(1)p 1475 4511 305 4 v 1475 4603 a(\()p Fl(n)h Fu(\000)f Fv(2\))1806 4364 y Fj( )1885 4534 y Fl(nw)2013 4549 y Fp(ii)2088 4534 y Fu(\000)2238 4410 y Fp(n)2187 4440 y Fj(X)2198 4650 y Fp(j)t Fo(=1)2348 4534 y Fl(w)2418 4549 y Fp(j)t(i)2500 4534 y Fu(\000)2650 4410 y Fp(n)2600 4440 y Fj(X)2610 4650 y Fp(j)t Fo(=1)2760 4534 y Fl(w)2830 4549 y Fp(ij)2891 4364 y Fj(!)330 4805 y Fv(W)-8 b(e)32 b(use)h(these)g(form)m(ulas)d(to)i(express)i(the)e (pro)5 b(jection)32 b Fl(P)2460 4820 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2652 4805 y Fv(\()p Fl(A)p Fv(\))f(of)h(a)f(matrix)g Fl(A)g Fv(in)m(to)h Fl(L)3708 4820 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)330 5022 y Fv(as)h(the)g(sum)f(of)g(ro)m(w,)h (column)f(and)g(diagonal)e(matrices)i(whose)i(en)m(tries)f(sum)f(to)g (zero.)44 b(De\014ne)p eop %%Page: 47 55 47 54 bop 2070 -210 a Fw(47)330 91 y Fv(the)33 b(follo)m(wing)d(op)s (erators)i(on)h(Mat)1653 106 y Fp(n)1700 91 y Fv(:)380 408 y Fl(Q)457 367 y Fo(1)497 408 y Fv(\()p Fl(A)p Fv(\))646 437 y Fp(ij)801 408 y Fv(:=)1098 340 y(1)p 941 385 363 4 v 941 476 a Fl(n)p Fv(\()p Fl(n)23 b Fu(\000)f Fv(2\))1331 237 y Fj( )1409 408 y Fv(\()p Fl(n)h Fu(\000)f Fv(1\))1782 283 y Fp(n)1731 313 y Fj(X)1731 526 y Fp(j)1764 507 y Fh(0)1785 526 y Fo(=1)1892 408 y Fl(a)1943 423 y Fp(ij)2000 404 y Fh(0)2049 408 y Fv(+)2198 283 y Fp(n)2147 313 y Fj(X)2147 526 y Fp(j)2180 507 y Fh(0)2201 526 y Fo(=1)2308 408 y Fl(a)2359 423 y Fp(j)2392 404 y Fh(0)2414 423 y Fp(i)2464 408 y Fu(\000)2648 283 y Fp(n)2597 313 y Fj(X)2564 526 y Fp(i)2588 507 y Fh(0)2610 526 y Fp(;j)2663 507 y Fh(0)2685 526 y Fo(=1)2791 408 y Fl(a)2842 423 y Fp(i)2866 404 y Fh(0)2889 423 y Fp(j)2922 404 y Fh(0)2970 408 y Fu(\000)h Fl(na)3179 423 y Fp(ii)3253 408 y Fv(+)3402 283 y Fp(n)3351 313 y Fj(X)3355 526 y Fp(i)3379 507 y Fh(0)3402 526 y Fo(=1)3512 408 y Fl(a)3563 423 y Fp(i)3587 404 y Fh(0)3610 423 y Fp(i)3634 404 y Fh(0)3660 237 y Fj(!)380 756 y Fl(Q)457 715 y Fo(2)497 756 y Fv(\()p Fl(A)p Fv(\))646 785 y Fp(ij)793 756 y Fv(:=)1091 689 y(1)p 934 733 V 934 824 a Fl(n)p Fv(\()p Fl(n)f Fu(\000)h Fv(2\))1323 586 y Fj( )1402 756 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\))1774 632 y Fp(n)1723 661 y Fj(X)1727 874 y Fp(i)1751 855 y Fh(0)1773 874 y Fo(=1)1884 756 y Fl(a)1935 771 y Fp(i)1959 752 y Fh(0)1982 771 y Fp(j)2040 756 y Fv(+)2189 632 y Fp(n)2138 661 y Fj(X)2142 874 y Fp(i)2166 855 y Fh(0)2188 874 y Fo(=1)2299 756 y Fl(a)2350 771 y Fp(j)t(i)2407 752 y Fh(0)2455 756 y Fu(\000)2639 632 y Fp(n)2588 661 y Fj(X)2555 874 y Fp(i)2579 855 y Fh(0)2601 874 y Fp(;j)2654 855 y Fh(0)2675 874 y Fo(=1)2782 756 y Fl(a)2833 771 y Fp(i)2857 752 y Fh(0)2880 771 y Fp(j)2913 752 y Fh(0)2961 756 y Fu(\000)f Fl(na)3169 771 y Fp(j)t(j)3261 756 y Fv(+)3409 632 y Fp(n)3359 661 y Fj(X)3362 874 y Fp(i)3386 855 y Fh(0)3409 874 y Fo(=1)3519 756 y Fl(a)3570 771 y Fp(i)3594 752 y Fh(0)3617 771 y Fp(i)3641 752 y Fh(0)3667 586 y Fj(!)380 1254 y Fl(Q)457 1213 y Fo(3)497 1254 y Fv(\()p Fl(A)p Fv(\))646 1283 y Fp(ij)789 930 y Fj(8)789 1020 y(>)789 1050 y(>)789 1080 y(>)789 1109 y(>)789 1139 y(<)789 1319 y(>)789 1349 y(>)789 1378 y(>)789 1408 y(>)789 1438 y(:)978 1052 y Fv(1)p 888 1097 229 4 v 888 1188 a Fl(n)g Fu(\000)h Fv(2)1143 949 y Fj( )1222 1120 y Fl(na)1331 1135 y Fp(ii)1406 1120 y Fu(\000)1556 995 y Fp(n)1506 1025 y Fj(X)1509 1238 y Fp(i)1533 1219 y Fh(0)1556 1238 y Fo(=1)1666 1120 y Fl(a)1717 1135 y Fp(i)1741 1116 y Fh(0)1764 1135 y Fp(i)1788 1116 y Fh(0)1836 1120 y Fu(\000)1987 995 y Fp(n)1937 1025 y Fj(X)1936 1238 y Fp(j)1969 1219 y Fh(0)1991 1238 y Fo(=1)2098 1120 y Fl(a)2149 1135 y Fp(ij)2206 1116 y Fh(0)2254 1120 y Fu(\000)2404 995 y Fp(n)2354 1025 y Fj(X)2353 1238 y Fp(j)2386 1219 y Fh(0)2408 1238 y Fo(=1)2515 1120 y Fl(a)2566 1135 y Fp(j)2599 1116 y Fh(0)2621 1135 y Fp(i)2671 1120 y Fv(+)2784 1052 y(2)p 2779 1097 59 4 v 2779 1188 a Fl(n)2948 995 y Fp(n)2897 1025 y Fj(X)2864 1238 y Fp(i)2888 1219 y Fh(0)2910 1238 y Fp(;j)2963 1219 y Fh(0)2985 1238 y Fo(=1)3091 1120 y Fl(a)3142 1135 y Fp(i)3166 1116 y Fh(0)3189 1135 y Fp(j)3222 1116 y Fh(0)3248 949 y Fj(!)3424 1120 y Fv(if)64 b Fl(i)28 b Fv(=)g Fl(j)878 1442 y Fv(0)2497 b(if)64 b Fl(i)28 b Fu(6)p Fv(=)g Fl(j)330 1715 y Fv(Then)k Fl(P)646 1730 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)866 1715 y Fv(=)c Fl(Q)1047 1679 y Fo(1)1106 1715 y Fv(+)19 b Fl(Q)1278 1679 y Fo(2)1337 1715 y Fv(+)g Fl(Q)1509 1679 y Fo(3)1580 1715 y Fv(since)32 b(this)f(op)s(erator)g(is)f(linear,)g(idemp)s(oten)m (t)h(and)g(has)h(range)330 1931 y Fl(L)396 1946 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)588 1931 y Fv(.)42 b(Similarly)-8 b(,)25 b(w)m(e)j(can)g(describ)s(e)h(the)f(pro)5 b(jection)27 b(from)f(Mat)2794 1946 y Fp(n)2858 1931 y Fu(\012)17 b Fv(Mat)3128 1946 y Fp(n)3203 1931 y Fv(to)27 b Fl(L)3383 1946 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)3587 1931 y Fu(\012)12 b Fv(Mat)3853 1946 y Fp(n)330 2148 y Fv(as)45 b(the)h(sum)f(of)f(three)i(op)s(erators)f Fl(Q)1768 2112 y Fo(1)1807 2148 y Fv(\()p Fl(C)7 b Fv(\),)48 b Fl(Q)2112 2112 y Fo(2)2152 2148 y Fv(\()p Fl(C)7 b Fv(\))45 b(and)g Fl(Q)2629 2112 y Fo(3)2669 2148 y Fv(\()p Fl(C)7 b Fv(\).)80 b(W)-8 b(e)46 b(use)g(the)f(shorthand)330 2365 y Fl(C)407 2324 y Fp(A)400 2389 y(k)r(l)498 2365 y Fv(=)693 2240 y Fp(n)642 2270 y Fj(X)609 2483 y Fp(i)633 2464 y Fh(0)655 2483 y Fp(;j)708 2464 y Fh(0)729 2483 y Fo(=1)836 2365 y Fl(c)878 2318 y Fp(i)902 2294 y Fh(0)924 2318 y Fp(j)957 2294 y Fh(0)878 2393 y Fp(k)r(l)1020 2365 y Fv(for)35 b(the)i(sum)g(of)f(all)e(en)m(tries)j(of)f(\()p Fl(c)2320 2318 y Fp(ij)2320 2393 y(k)r(l)2384 2365 y Fv(\))g(giv)m(en)h(\014xed)g Fl(k)s(;)17 b(l)39 b Fv(and)d Fl(C)3392 2324 y Fp(D)3385 2389 y(k)r(l)3490 2365 y Fv(=)3651 2240 y Fp(n)3601 2270 y Fj(X)3649 2483 y Fp(i)3673 2464 y Fh(0)3761 2365 y Fl(c)3803 2324 y Fp(i)3827 2300 y Fh(0)3849 2324 y Fp(i)3873 2300 y Fh(0)3803 2389 y Fp(k)r(l)330 2605 y Fv(for)c(the)h(sum)g(of)f (the)h(diagonal)d(en)m(tries)j(of)f(\()p Fl(c)2026 2558 y Fp(ij)2026 2633 y(k)r(l)2091 2605 y Fv(\))g(giv)m(en)h(\014xed)g Fl(k)s(;)17 b(l)r Fv(:)601 2921 y Fl(Q)678 2880 y Fo(1)718 2921 y Fv(\()p Fl(C)7 b Fv(\))870 2864 y Fp(ij)870 2951 y(k)r(l)1021 2921 y Fv(:=)1319 2854 y(1)p 1161 2899 363 4 v 1161 2990 a Fl(n)p Fv(\()p Fl(n)23 b Fu(\000)g Fv(2\))1551 2751 y Fj( )1630 2921 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\))2002 2797 y Fp(n)1952 2827 y Fj(X)1952 3039 y Fp(j)1985 3021 y Fh(0)2006 3039 y Fo(=1)2113 2921 y Fl(c)2155 2874 y Fp(ij)2212 2851 y Fh(0)2155 2949 y Fp(k)r(l)2260 2921 y Fv(+)2409 2797 y Fp(n)2358 2827 y Fj(X)2358 3039 y Fp(j)2391 3021 y Fh(0)2412 3039 y Fo(=1)2519 2921 y Fl(c)2561 2874 y Fp(j)2594 2851 y Fh(0)2616 2874 y Fp(i)2561 2949 y(k)r(l)2666 2921 y Fu(\000)g Fl(nc)2866 2880 y Fp(ii)2866 2946 y(k)r(l)2953 2921 y Fu(\000)f Fl(C)3129 2880 y Fp(A)3122 2946 y(k)r(l)3209 2921 y Fv(+)g Fl(C)3384 2880 y Fp(D)3377 2946 y(k)r(l)3448 2751 y Fj(!)601 3270 y Fl(Q)678 3229 y Fo(2)718 3270 y Fv(\()p Fl(C)7 b Fv(\))870 3212 y Fp(ij)870 3299 y(k)r(l)1020 3270 y Fv(:=)1317 3202 y(1)p 1160 3247 V 1160 3338 a Fl(n)p Fv(\()p Fl(n)23 b Fu(\000)f Fv(2\))1550 3100 y Fj( )1629 3270 y Fv(\()p Fl(n)g Fu(\000)h Fv(1\))2000 3145 y Fp(n)1950 3175 y Fj(X)1954 3388 y Fp(i)1978 3369 y Fh(0)2000 3388 y Fo(=1)2110 3270 y Fl(c)2152 3223 y Fp(i)2176 3199 y Fh(0)2199 3223 y Fp(j)2152 3298 y(k)r(l)2257 3270 y Fv(+)2406 3145 y Fp(n)2355 3175 y Fj(X)2359 3388 y Fp(i)2383 3369 y Fh(0)2405 3388 y Fo(=1)2516 3270 y Fl(c)2558 3223 y Fp(j)t(i)2615 3199 y Fh(0)2558 3298 y Fp(k)r(l)2663 3270 y Fu(\000)g Fl(nc)2863 3223 y Fp(j)t(j)2863 3298 y(k)r(l)2954 3270 y Fu(\000)f Fl(C)3130 3229 y Fp(A)3123 3295 y(k)r(l)3210 3270 y Fv(+)g Fl(C)3385 3229 y Fp(D)3378 3295 y(k)r(l)3449 3100 y Fj(!)601 3755 y Fl(Q)678 3714 y Fo(3)718 3755 y Fv(\()p Fl(C)7 b Fv(\))870 3697 y Fp(ij)870 3784 y(k)r(l)1018 3431 y Fj(8)1018 3521 y(>)1018 3551 y(>)1018 3580 y(>)1018 3610 y(>)1018 3640 y(<)1018 3820 y(>)1018 3849 y(>)1018 3879 y(>)1018 3909 y(>)1018 3939 y(:)1207 3553 y Fv(1)p 1117 3598 229 4 v 1117 3689 a Fl(n)22 b Fu(\000)h Fv(2)1372 3450 y Fj( )1451 3620 y Fl(nc)1551 3579 y Fp(ii)1551 3645 y(k)r(l)1638 3620 y Fu(\000)1788 3496 y Fp(n)1738 3526 y Fj(X)1737 3738 y Fp(j)1770 3720 y Fh(0)1792 3738 y Fo(=1)1899 3620 y Fl(c)1941 3573 y Fp(ij)1998 3550 y Fh(0)1941 3648 y Fp(k)r(l)2046 3620 y Fu(\000)2196 3496 y Fp(n)2146 3526 y Fj(X)2145 3738 y Fp(j)2178 3720 y Fh(0)2200 3738 y Fo(=1)2307 3620 y Fl(c)2349 3573 y Fp(j)2382 3550 y Fh(0)2404 3573 y Fp(i)2349 3648 y(k)r(l)2454 3620 y Fu(\000)g Fl(C)2631 3579 y Fp(D)2624 3645 y(k)r(l)2717 3620 y Fv(+)2829 3553 y(2)p 2825 3598 59 4 v 2825 3689 a Fl(n)2893 3620 y(C)2970 3579 y Fp(A)2963 3645 y(k)r(l)3027 3450 y Fj(!)3204 3620 y Fv(if)64 b Fl(i)27 b Fv(=)h Fl(j)1107 3943 y Fv(0)2048 b(if)64 b Fl(i)27 b Fu(6)p Fv(=)h Fl(j)471 4206 y Fv(Since)h Fl(f)770 4221 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)991 4206 y Fv(is)f(linear,)g(w)m(e)h(can)g(break)h(it)d(up)i(as)g(a)g(sum)f (of)g(functions)h(on)g(the)g(three)g(pieces)330 4423 y(of)j Fl(P)504 4438 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)696 4423 y Fv(\()p Fl(C)7 b Fv(\):)1449 4639 y Fl(f)1497 4654 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1717 4639 y Fv(=)27 b Fl(f)1879 4598 y Fo(1)1868 4664 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2083 4639 y Fv(+)22 b Fl(f)2240 4598 y Fo(2)2229 4664 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2443 4639 y Fv(+)g Fl(f)2600 4598 y Fo(3)2589 4664 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)p eop %%Page: 48 56 48 55 bop 2070 -210 a Fw(48)330 91 y Fv(where)34 b(w)m(e)f(ha)m(v)m(e:) 413 433 y Fl(f)472 391 y Fo(1)461 457 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)653 433 y Fv(\()p Fl(\033)t Fv(\))28 b(:=)441 581 y Fp(n)391 610 y Fj(X)380 820 y Fp(i;j)t Fo(=1)730 638 y Fv(1)p 572 682 363 4 v 572 773 a Fl(n)p Fv(\()p Fl(n)23 b Fu(\000)g Fv(2\))962 535 y Fj( )1041 705 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\))1413 581 y Fp(n)1363 610 y Fj(X)1363 823 y Fp(j)1396 804 y Fh(0)1417 823 y Fo(=1)1524 705 y Fl(c)1566 658 y Fp(ij)1623 634 y Fh(0)1566 737 y Fp(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))1843 705 y Fv(+)1992 581 y Fp(n)1942 610 y Fj(X)1941 823 y Fp(j)1974 804 y Fh(0)1996 823 y Fo(=1)2103 705 y Fl(c)2145 658 y Fp(j)2178 634 y Fh(0)2200 658 y Fp(i)2145 737 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))2423 705 y Fu(\000)f Fl(nc)2622 664 y Fp(ii)2622 730 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))2901 705 y Fu(\000)g Fl(C)3077 664 y Fp(A)3070 730 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))3348 705 y Fv(+)g Fl(C)3523 664 y Fp(D)3516 730 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))3771 535 y Fj(!)358 1054 y Fv(=)512 929 y Fp(n)461 959 y Fj(X)476 1169 y Fp(i)p Fo(=1)789 986 y Fv(1)p 632 1031 V 632 1122 a Fl(n)p Fv(\()p Fl(n)g Fu(\000)h Fv(2\))1021 883 y Fj( )1100 1054 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\))1496 929 y Fp(n)1446 959 y Fj(X)1422 1172 y Fp(j;j)1504 1153 y Fh(0)1524 1172 y Fo(=1)1631 1054 y Fl(c)1673 1006 y Fp(ij)1730 983 y Fh(0)1673 1085 y Fp(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(j)1853 1054 y Fv(+)2026 929 y Fp(n)1976 959 y Fj(X)1951 1172 y Fp(j;j)2033 1153 y Fh(0)2054 1172 y Fo(=1)2161 1054 y Fl(c)2203 1006 y Fp(j)2236 983 y Fh(0)2257 1006 y Fp(i)2203 1085 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(j)2383 1054 y Fu(\000)g Fl(n)2608 929 y Fp(n)2557 959 y Fj(X)2568 1169 y Fp(j)t Fo(=1)2718 1054 y Fl(c)2760 1012 y Fp(ii)2760 1078 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(j)2940 1054 y Fu(\000)3090 929 y Fp(n)3040 959 y Fj(X)3050 1169 y Fp(j)t Fo(=1)3200 1054 y Fl(C)3277 1012 y Fp(A)3270 1078 y(ij)3356 1054 y Fv(+)3505 929 y Fp(n)3454 959 y Fj(X)3465 1169 y Fp(j)t Fo(=1)3615 1054 y Fl(C)3692 1012 y Fp(D)3685 1078 y(ij)3756 883 y Fj(!)413 1579 y Fl(f)472 1537 y Fo(2)461 1603 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)653 1579 y Fv(\()p Fl(\033)t Fv(\))28 b(:=)442 1727 y Fp(n)392 1756 y Fj(X)381 1966 y Fp(i;j)t Fo(=1)731 1784 y Fv(1)p 573 1828 V 573 1919 a Fl(n)p Fv(\()p Fl(n)23 b Fu(\000)g Fv(2\))963 1681 y Fj( )1042 1851 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\))1414 1727 y Fp(n)1363 1756 y Fj(X)1367 1969 y Fp(i)1391 1950 y Fh(0)1413 1969 y Fo(=1)1524 1851 y Fl(c)1566 1804 y Fp(i)1590 1780 y Fh(0)1612 1804 y Fp(j)1566 1883 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))1843 1851 y Fv(+)1992 1727 y Fp(n)1941 1756 y Fj(X)1945 1969 y Fp(i)1969 1950 y Fh(0)1991 1969 y Fo(=1)2102 1851 y Fl(c)2144 1804 y Fp(j)t(i)2201 1780 y Fh(0)2144 1883 y Fp(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))2422 1851 y Fu(\000)f Fl(nc)2621 1804 y Fp(j)t(j)2621 1883 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))2900 1851 y Fu(\000)g Fl(C)3076 1810 y Fp(A)3069 1876 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))3347 1851 y Fv(+)g Fl(C)3522 1810 y Fp(D)3515 1876 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))3770 1681 y Fj(!)358 2197 y Fv(=)512 2072 y Fp(n)461 2102 y Fj(X)472 2312 y Fp(j)t Fo(=1)789 2129 y Fv(1)p 632 2174 V 632 2265 a Fl(n)p Fv(\()p Fl(n)g Fu(\000)h Fv(2\))1021 2026 y Fj( )1100 2197 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\))1490 2072 y Fp(n)1439 2102 y Fj(X)1422 2315 y Fp(i;i)1490 2296 y Fh(0)1511 2315 y Fo(=1)1618 2197 y Fl(c)1660 2150 y Fp(i)1684 2126 y Fh(0)1706 2150 y Fp(j)1660 2228 y(i\033)r Fo(\()p Fp(j)t Fo(\))1840 2197 y Fv(+)2007 2072 y Fp(n)1956 2102 y Fj(X)1938 2315 y Fp(i;i)2006 2296 y Fh(0)2028 2315 y Fo(=1)2135 2197 y Fl(c)2177 2150 y Fp(j)t(i)2234 2126 y Fh(0)2177 2228 y Fp(i\033)r Fo(\()p Fp(j)t Fo(\))2357 2197 y Fu(\000)g Fl(n)2582 2072 y Fp(n)2532 2102 y Fj(X)2547 2312 y Fp(i)p Fo(=1)2692 2197 y Fl(c)2734 2150 y Fp(j)t(j)2734 2228 y(i\033)r Fo(\()p Fp(j)t Fo(\))2915 2197 y Fu(\000)3065 2072 y Fp(n)3014 2102 y Fj(X)3029 2312 y Fp(i)p Fo(=1)3175 2197 y Fl(C)3252 2156 y Fp(A)3245 2221 y(ij)3331 2197 y Fv(+)3479 2072 y Fp(n)3429 2102 y Fj(X)3444 2312 y Fp(i)p Fo(=1)3589 2197 y Fl(C)3666 2156 y Fp(D)3659 2221 y(ij)3730 2026 y Fj(!)413 2722 y Fl(f)472 2681 y Fo(3)461 2746 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)653 2722 y Fv(\()p Fl(\033)t Fv(\))28 b(:=)744 2870 y Fp(n)694 2900 y Fj(X)708 3110 y Fp(i)p Fo(=1)992 2927 y Fv(1)p 864 2971 305 4 v 864 3063 a(\()p Fl(n)22 b Fu(\000)h Fv(2\))1195 2824 y Fj( )1274 2994 y Fl(nc)1374 2953 y Fp(ii)1374 3019 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(i)p Fo(\))1644 2994 y Fu(\000)1794 2870 y Fp(n)1744 2900 y Fj(X)1743 3112 y Fp(j)1776 3093 y Fh(0)1798 3112 y Fo(=1)1905 2994 y Fl(c)1947 2947 y Fp(ij)2004 2924 y Fh(0)1947 3026 y Fp(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(i)p Fo(\))2216 2994 y Fu(\000)2367 2870 y Fp(n)2317 2900 y Fj(X)2316 3112 y Fp(j)2349 3093 y Fh(0)2371 3112 y Fo(=1)2478 2994 y Fl(c)2520 2947 y Fp(j)2553 2924 y Fh(0)2574 2947 y Fp(i)2520 3026 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(i)p Fo(\))2789 2994 y Fu(\000)g Fl(C)2966 2953 y Fp(D)2959 3019 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(i)p Fo(\))3228 2994 y Fv(+)3341 2927 y(2)p 3336 2971 59 4 v 3336 3063 a Fl(n)3404 2994 y(C)3481 2953 y Fp(A)3474 3019 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(i)p Fo(\))3721 2824 y Fj(!)330 3322 y Fv(Note)41 b(that)g(in)g(the)h(expressions)h(for)d Fl(f)1827 3286 y Fo(1)1816 3347 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2050 3322 y Fv(and)h Fl(f)2307 3286 y Fo(2)2296 3347 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2530 3322 y Fv(w)m(e)h(are)f(able)g(to)g(re-index)g Fl(j)47 b Fv(and)42 b Fl(i)330 3539 y Fv(resp)s(ectiv)m(ely)f(since)e(the)h(sum)g(is)f(o)m (v)m(er)h(the)g(en)m(tire)g(set)g Fu(f)p Fv(1)p Fl(;)17 b(:::;)g(n)p Fu(g)p Fv(.)64 b(Similarly)-8 b(,)37 b(w)m(e)j(note)g (that)392 3631 y Fp(n)341 3661 y Fj(X)330 3871 y Fp(i;j)t Fo(=1)513 3756 y Fl(C)590 3714 y Fp(A)583 3780 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))871 3756 y Fv(and)1122 3631 y Fp(n)1072 3661 y Fj(X)1061 3871 y Fp(i;j)t Fo(=1)1243 3756 y Fl(C)1320 3714 y Fp(D)1313 3780 y(\033)r Fo(\()p Fp(i)p Fo(\))p Fp(\033)r Fo(\()p Fp(j)t Fo(\))1601 3756 y Fv(do)33 b(not)f(dep)s(end)i(on)e Fl(\033)t Fv(.)471 3972 y(No)m(w)h(consider)g(the)g(matrix)e Fl(D)g Fv(=)c(\()p Fl(d)1862 3987 y Fp(ik)1928 3972 y Fv(\))33 b(giv)m(en)g(b)m(y:)519 4289 y Fl(d)570 4304 y Fp(ik)719 4289 y Fv(:=)1017 4221 y(1)p 860 4266 363 4 v 860 4357 a Fl(n)p Fv(\()p Fl(n)22 b Fu(\000)h Fv(2\))1232 4118 y Fj( )1394 4289 y Fv(\()p Fl(n)g Fu(\000)f Fv(1\))1774 4164 y Fp(n)1724 4194 y Fj(X)1716 4406 y Fp(j;l)q Fo(=1)1892 4289 y Fl(c)1934 4241 y Fp(ij)1934 4316 y(k)r(l)2021 4289 y Fv(+)2177 4164 y Fp(n)2127 4194 y Fj(X)2119 4406 y Fp(j;l)q Fo(=1)2295 4289 y Fl(c)2337 4241 y Fp(j)t(i)2337 4316 y(k)r(l)2424 4289 y Fu(\000)h Fl(n)2649 4164 y Fp(n)2599 4194 y Fj(X)2615 4406 y Fp(l)q Fo(=1)2759 4289 y Fl(c)2801 4247 y Fp(ii)2801 4313 y(k)r(l)2888 4289 y Fv(+)f(\()p Fl(n)g Fu(\000)h Fv(1\))3366 4164 y Fp(n)3315 4194 y Fj(X)3308 4406 y Fp(j;l)q Fo(=1)3484 4289 y Fl(c)3526 4241 y Fp(j)t(i)3526 4316 y(l)q(k)1394 4636 y Fv(+)1545 4512 y Fp(n)1495 4542 y Fj(X)1487 4754 y Fp(j;l)q Fo(=1)1663 4636 y Fl(c)1705 4589 y Fp(ij)1705 4664 y(l)q(k)1792 4636 y Fu(\000)g Fl(n)2017 4512 y Fp(n)1966 4542 y Fj(X)1982 4754 y Fp(l)q Fo(=1)2127 4636 y Fl(c)2169 4595 y Fp(ii)2169 4661 y(l)q(k)2256 4636 y Fv(+)f Fl(n)2412 4595 y Fo(2)2451 4636 y Fl(c)2493 4595 y Fp(ii)2493 4661 y(k)r(k)2597 4636 y Fu(\000)g Fl(n)2822 4512 y Fp(n)2771 4542 y Fj(X)2782 4752 y Fp(j)t Fo(=1)2932 4636 y Fl(c)2974 4589 y Fp(ij)2974 4664 y(k)r(k)3077 4636 y Fu(\000)h Fl(n)3302 4512 y Fp(n)3251 4542 y Fj(X)3262 4752 y Fp(j)t Fo(=1)3412 4636 y Fl(c)3454 4589 y Fp(j)t(i)3454 4664 y(k)r(k)3633 4466 y Fj(!)732 4984 y Fv(=)1003 4917 y(1)p 846 4962 V 846 5053 a Fl(n)p Fv(\()p Fl(n)g Fu(\000)f Fv(2\))1219 4814 y Fj( )1394 4984 y Fv(\()p Fl(n)1490 4943 y Fo(2)1552 4984 y Fu(\000)h Fv(2)p Fl(n)p Fv(\))p Fl(c)1839 4943 y Fp(ii)1839 5009 y(k)r(k)1942 4984 y Fv(+)2040 4890 y Fj(X)2056 5102 y Fp(j)t Ff(6)p Fo(=)p Fp(i)2200 4890 y Fj(X)2215 5102 y Fp(l)q Ff(6)p Fo(=)p Fp(k)2344 4874 y Fj(\020)2404 4984 y Fv(\()p Fl(n)f Fu(\000)h Fv(1\)\()p Fl(c)2789 4937 y Fp(ij)2789 5012 y(k)r(l)2875 4984 y Fv(+)f Fl(c)3015 4937 y Fp(j)t(i)3015 5012 y(l)q(k)3080 4984 y Fv(\))g(+)g Fl(c)3280 4937 y Fp(ij)3280 5012 y(l)q(k)3366 4984 y Fv(+)g Fl(c)3506 4937 y Fp(j)t(i)3506 5012 y(k)r(l)3571 4874 y Fj(\021)3630 4814 y(!)330 5301 y Fv(Then)30 b Fl(D)i Fv(de\014nes)e(a)f(linear)e(assignmen)m(t)i(problem)e(\(see)j (De\014nition)d(1.1.4\))i(giv)m(en)g(b)m(y)h(the)f(func-)p eop %%Page: 49 57 49 56 bop 2070 -210 a Fw(49)330 91 y Fv(tion:)1773 408 y Fl(h)p Fv(\()p Fl(\033)t Fv(\))28 b(=)2146 283 y Fp(n)2096 313 y Fj(X)2110 523 y Fp(i)p Fo(=1)2256 408 y Fl(d)2307 423 y Fp(i\033)r Fo(\()p Fp(i)p Fo(\))3699 408 y Fv(\(3.3\))330 724 y(W)-8 b(e)33 b(c)m(hec)m(k)i(that)d(if)g(w)m(e)h(set:)612 1040 y Fl(d)83 b Fv(=)864 973 y(1)p 859 1017 59 4 v 859 1109 a Fl(n)1009 916 y Fp(n)958 946 y Fj(X)944 1158 y Fp(i;k)r Fo(=1)1133 1040 y Fl(d)1184 1055 y Fp(ik)1278 1040 y Fv(=)1548 973 y(1)p 1391 1017 363 4 v 1391 1109 a Fl(n)p Fv(\()p Fl(n)23 b Fu(\000)g Fv(2\))1781 870 y Fj( )1921 916 y Fp(n)1871 946 y Fj(X)1860 1156 y Fp(i;j)t Fo(=1)2026 1040 y Fv(\(2)p Fl(C)2190 999 y Fp(D)2183 1065 y(ij)2276 1040 y Fu(\000)f Fv(2)p Fl(C)2501 999 y Fp(A)2494 1065 y(ij)2558 1040 y Fv(\))g(+)2767 916 y Fp(n)2716 946 y Fj(X)2731 1156 y Fp(i)p Fo(=1)2860 1040 y Fv(\(2)p Fl(C)3024 999 y Fp(A)3017 1065 y(ij)3103 1040 y Fu(\000)g Fl(nC)3337 999 y Fp(D)3330 1065 y(ij)3402 1040 y Fv(\))3440 870 y Fj(!)1411 1377 y Fv(=)1681 1310 y(1)p 1524 1354 V 1524 1445 a Fl(n)p Fv(\()p Fl(n)h Fu(\000)f Fv(2\))1914 1282 y Fj(X)1930 1494 y Fp(j)t Ff(6)p Fo(=)p Fp(i)2074 1282 y Fj(X)2089 1494 y Fp(l)q Ff(6)p Fo(=)p Fp(k)2235 1377 y Fl(c)2277 1330 y Fp(ij)2277 1405 y(k)r(l)2363 1377 y Fu(\000)2478 1310 y Fv(1)p 2473 1354 59 4 v 2473 1445 a Fl(n)2623 1252 y Fp(n)2572 1282 y Fj(X)2558 1494 y Fp(i;k)r Fo(=1)2747 1377 y Fl(c)2789 1336 y Fp(ii)2789 1402 y(k)r(k)330 1693 y Fv(then)33 b(w)m(e)h(ha)m(v)m(e:)447 2010 y Fl(h)p Fv(\()p Fl(\033)t Fv(\))28 b(=)g Fl(f)829 1968 y Fo(1)818 2034 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1010 2010 y Fv(\()p Fl(\033)t Fv(\))22 b(+)g Fl(f)1324 1968 y Fo(2)1313 2034 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1505 2010 y Fv(\()p Fl(\033)t Fv(\))g(+)g Fl(f)1819 1968 y Fo(3)1808 2034 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2000 2010 y Fv(\()p Fl(\033)t Fv(\))g(+)g Fl(d)27 b Fv(=)h Fl(f)2485 2025 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2677 2010 y Fv(\()p Fl(\033)t Fv(\))22 b(+)g Fl(d)227 b Fv(for)32 b(all)f Fl(\033)h Fu(2)c Fl(S)3736 2025 y Fp(n)330 2326 y Fv(So)e(the)h(problem)f(of)g(c)m(hec)m(king)i(whether)g Fl(f)1903 2341 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2095 2326 y Fv(\()p Fl(\033)t Fv(\))f(=)h(0)e(ev)m(erywhere)k(reduces)e(to)e (the)h(problem)330 2543 y(of)e(c)m(hec)m(king)i(whether)g(\(3.3\))e(is) g(constan)m(t.)42 b(W)-8 b(e)26 b(remark)g(that)f(this)g(is)g(a)h (problem)e(of)h(computing)330 2759 y(the)33 b(cen)m(tral)f(pro)5 b(jection)33 b(of)f(a)g Fk(LAP)i Fv(\(see)f(the)g(discussion)g(in)f (Section)g(4.1.1\).)471 2976 y(W)-8 b(e)49 b(can)g(state)h(fairly)c (simple)i(necessary)j(and)e(su\016cien)m(t)h(conditions)d(for)i Fl(h)p Fv(\()p Fl(\033)t Fv(\))f(to)h(b)s(e)330 3193 y(constan)m(t)37 b(for)f(all)e Fl(\033)k Fu(2)c Fl(S)1272 3208 y Fp(n)1356 3193 y Fv(in)h(terms)h(of)g(the)h(matrix)d Fl(D)s Fv(.)55 b(They)37 b(are)g(exactly)f(that)g Fl(D)j Fv(can)e(b)s(e)330 3409 y(written)i(a)g(sum)g(of)g(a)g(ro)m(w)g(matrix) f(and)h(a)g(column)f(matrix,)i(and)f(has)g(en)m(tries)h(that)f(sum)g (to)330 3626 y(zero.)44 b(That)33 b(is)f(for)g(some)g(pair)g(of)g(v)m (ectors)i Fl(a;)17 b(b)28 b Fu(2)g Fi(R)2276 3590 y Fp(n)2362 3626 y Fv(suc)m(h)34 b(that)2844 3502 y Fp(n)2793 3531 y Fj(X)2808 3741 y Fp(i)p Fo(=1)2937 3626 y Fv(\()p Fl(a)3026 3641 y Fp(i)3076 3626 y Fv(+)22 b Fl(b)3215 3641 y Fp(i)3244 3626 y Fv(\))28 b(=)f(0,)33 b(w)m(e)g(ha)m(v)m(e:)852 4318 y Fl(D)d Fv(=)1067 3878 y Fj(0)1067 4054 y(B)1067 4113 y(B)1067 4173 y(B)1067 4233 y(B)1067 4293 y(B)1067 4352 y(B)1067 4412 y(B)1067 4472 y(B)1067 4536 y(@)1158 4011 y Fl(a)1209 4026 y Fo(1)1271 4011 y Fv(+)22 b Fl(b)1410 4026 y Fo(1)1540 4011 y Fl(a)1591 4026 y Fo(1)1653 4011 y Fv(+)g Fl(b)1792 4026 y Fo(2)1919 4011 y Fl(:)17 b(:)g(:)103 b(a)2188 4026 y Fo(1)2249 4011 y Fv(+)22 b Fl(b)2388 4026 y Fp(n)1158 4228 y Fl(a)1209 4243 y Fo(2)1271 4228 y Fv(+)g Fl(b)1410 4243 y Fo(1)1540 4228 y Fl(a)1591 4243 y Fo(2)1653 4228 y Fv(+)g Fl(b)1792 4243 y Fo(2)1919 4228 y Fl(:)17 b(:)g(:)103 b(a)2188 4243 y Fo(2)2249 4228 y Fv(+)22 b Fl(b)2388 4243 y Fp(n)1238 4444 y Fl(:)17 b(:)g(:)267 b(:)17 b(:)g(:)184 b(:)17 b(:)g(:)195 b(:)17 b(:)g(:)1154 4661 y(a)1205 4676 y Fp(n)1274 4661 y Fv(+)22 b Fl(b)1413 4676 y Fo(1)1536 4661 y Fl(a)1587 4676 y Fp(n)1657 4661 y Fv(+)g Fl(b)1796 4676 y Fo(2)1919 4661 y Fl(:)17 b(:)g(:)99 b(a)2184 4676 y Fp(n)2253 4661 y Fv(+)22 b Fl(b)2392 4676 y Fp(n)2440 3878 y Fj(1)2440 4054 y(C)2440 4113 y(C)2440 4173 y(C)2440 4233 y(C)2440 4293 y(C)2440 4352 y(C)2440 4412 y(C)2440 4472 y(C)2440 4536 y(A)2543 4318 y Fl(;)212 b(d)2833 4333 y Fp(ij)2921 4318 y Fv(=)28 b(\()p Fl(a)3114 4333 y Fp(i)3164 4318 y Fv(+)22 b Fl(b)3303 4333 y Fp(j)3340 4318 y Fv(\))330 4903 y(If)44 b Fl(D)j Fv(has)e(this)f(form,)i(then)f(it)f(is)g(clear)f (that)i(the)f(ob)5 b(jectiv)m(e)46 b(v)-5 b(alue)44 b(will)e(b)s(e)i (the)h(constan)m(t)381 4995 y Fp(n)330 5025 y Fj(X)345 5235 y Fp(i)p Fo(=1)474 5120 y Fv(\()p Fl(a)563 5135 y Fp(i)615 5120 y Fv(+)23 b Fl(b)755 5135 y Fp(i)784 5120 y Fv(\))35 b(for)f(all)f Fl(\033)i Fu(2)d Fl(S)1394 5135 y Fp(n)1441 5120 y Fv(.)51 b(An)m(y)35 b(t)m(w)m(o)h(v)m(ectors)g Fl(a)f Fv(and)g Fl(b)g Fv(de\014ne)i(suc)m(h)f(a)e Fl(D)k Fv(\(and,)d(without)330 5336 y(loss)d(of)g(generalit)m(y)-8 b(,)32 b(w)m(e)i(can)f(c)m(ho)s(ose)g Fl(a)1778 5351 y Fo(1)1845 5336 y Fv(=)28 b(0\).)p eop %%Page: 50 58 50 57 bop 2070 -210 a Fw(50)471 91 y Fv(Consider)23 b(for)e Fl(i;)c(k)31 b Fu(\025)d Fv(2)22 b(t)m(w)m(o)h(p)s(erm)m(utations,)g (with)f Fl(\033)2397 106 y Fo(1)2459 91 y Fv(satisfying)f Fl(\033)2938 106 y Fo(1)2978 91 y Fv(\(1\))27 b(=)h(1)21 b(and)i Fl(\033)3539 106 y Fo(1)3578 91 y Fv(\()p Fl(i)p Fv(\))28 b(=)g Fl(k)s Fv(,)330 308 y Fl(\033)385 323 y Fo(2)457 308 y Fv(satisfying)k Fl(\033)947 323 y Fo(2)987 308 y Fv(\(1\))27 b(=)h Fl(k)35 b Fv(and)e Fl(\033)1574 323 y Fo(2)1614 308 y Fv(\()p Fl(i)p Fv(\))28 b(=)f(1,)33 b(and)f Fl(\033)2207 323 y Fo(1)2247 308 y Fv(\()p Fl(j)6 b Fv(\))28 b(=)f Fl(\033)2555 323 y Fo(2)2595 308 y Fv(\()p Fl(j)6 b Fv(\))33 b(for)f(all)e Fl(j)k Fu(6)p Fv(=)27 b(1)p Fl(;)17 b(i)p Fv(.)43 b(Then:)1336 602 y Fl(h)p Fv(\()p Fl(\033)1485 617 y Fo(1)1525 602 y Fv(\))22 b Fu(\000)h Fl(d)1736 617 y Fo(11)1832 602 y Fu(\000)g Fl(d)1983 617 y Fp(ik)2077 602 y Fv(=)28 b Fl(h)p Fv(\()p Fl(\033)2330 617 y Fo(2)2370 602 y Fv(\))22 b Fu(\000)g Fl(d)2580 617 y Fo(1)p Fp(k)2680 602 y Fu(\000)h Fl(d)2831 617 y Fp(i)p Fo(1)330 896 y Fv(F)-8 b(or)32 b Fl(h)g Fv(to)h(b)s(e)f(constan)m(t)i(on)e(all)f(p)s(erm)m(utations,)h(w)m(e)h (ha)m(v)m(e)h Fl(h)p Fv(\()p Fl(\033)2651 911 y Fo(1)2691 896 y Fv(\))28 b(=)f Fl(h)p Fv(\()p Fl(\033)3009 911 y Fo(2)3049 896 y Fv(\),)33 b(hence:)1274 1190 y Fl(d)1325 1205 y Fo(11)1422 1190 y Fv(+)22 b Fl(d)1571 1205 y Fp(ik)1665 1190 y Fv(=)27 b Fl(d)1819 1205 y Fo(1)p Fp(k)1919 1190 y Fv(+)22 b Fl(d)2068 1205 y Fp(i)p Fo(1)2359 1190 y Fv(for)32 b(all)e Fl(i;)17 b(k)31 b Fu(\025)d Fv(2)743 b(\(3.4\))330 1484 y(These)35 b(\()p Fl(n)22 b Fu(\000)h Fv(1\))918 1448 y Fo(2)990 1484 y Fv(conditions)32 b(determine)g Fl(D)k Fv(giv)m(en)d(its)f(initial)e(ro)m(w)j(and)g(column)f(\(ie.)43 b(v)m(ectors)330 1700 y Fl(a)33 b Fv(and)f Fl(b)h Fv(ab)s(o)m(v)m(e\),) h(so)f(an)m(y)g Fl(D)i Fv(yielding)c(a)h(constan)m(t)i Fl(h)e 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b(can)g(p)s(erform)f(a)g (similar)e(analysis)i(for)g(the)h(c)m(haracter)h Fl(\037)2832 3329 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)3079 3314 y Fv(.)68 b(W)-8 b(e)41 b(calculate)e(the)330 3531 y(pro)5 b(jection)32 b Fl(P)856 3546 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)1103 3531 y Fv(\()p Fl(C)7 b Fv(\))32 b(of)h Fl(C)39 b Fv(in)m(to)32 b Fl(L)1773 3494 y Fo(2)1773 3555 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)2042 3531 y Fu(\012)23 b Fv(Mat)2318 3546 y Fp(n)2365 3531 y Fv(:)494 3824 y Fl(P)557 3839 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)803 3824 y Fv(\()p Fl(C)7 b Fv(\))956 3776 y Fp(ij)956 3856 y(k)r(l)1048 3824 y Fv(=)1162 3757 y(1)p 1162 3802 49 4 v 1162 3893 a(2)1237 3654 y Fj( )1316 3824 y Fl(c)1358 3777 y Fp(ij)1358 3852 y(k)r(l)1445 3824 y Fu(\000)22 b Fl(c)1586 3777 y Fp(j)t(i)1586 3852 y(k)r(l)1673 3824 y Fu(\000)1787 3757 y Fv(1)p 1783 3802 59 4 v 1783 3893 a Fl(n)1918 3700 y Fp(n)1867 3730 y Fj(X)1871 3942 y Fp(i)1895 3924 y 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Fv(\))26 b Fu(\025)j Fl(g)t Fv(\()p Fl(\033)t Fv(\))i(are)i(written)f(as)460 5380 y Fl(\013)522 5395 y Fo(1)562 5380 y Fv(\()p Fl(n)22 b Fu(\000)h Fv(1\))e(+)h Fl(\013)1048 5395 y Fo(2)1088 5380 y Fv(\()p Fl(n)1184 5339 y Fo(2)1246 5380 y Fu(\000)g Fv(2\))g(+)g Fl(\013)1614 5395 y Fo(3)1681 5380 y Fu(\025)28 b Fl(\013)1848 5395 y Fo(1)1888 5299 y Fj(\000)1934 5380 y Fl(p)p Fv(\()p Fl(\033)t Fv(\))22 b Fu(\000)g Fv(1)2288 5299 y Fj(\001)2356 5380 y Fv(+)g Fl(\013)2516 5395 y Fo(2)2555 5299 y Fj(\000)2601 5380 y Fl(p)2650 5339 y Fo(2)2689 5380 y Fv(\()p Fl(\033)t Fv(\))g Fu(\000)h Fv(2)2995 5299 y Fj(\001)3063 5380 y Fv(+)f Fl(\013)3223 5395 y Fo(3)3262 5299 y Fj(\000)3308 5380 y Fv(1)g Fu(\000)g Fv(2)p Fl(t)p Fv(\()p Fl(\033)t Fv(\))3697 5299 y Fj(\001)3743 5380 y Fl(;)p eop %%Page: 54 62 54 61 bop 2070 -210 a Fw(54)330 91 y Fv(whic)m(h)33 b(for)f Fl(\033)g Fu(6)p Fv(=)27 b Fl(")33 b Fv(are)f(equiv)-5 b(alen)m(t)32 b(to)1320 408 y Fl(\013)1382 423 y Fo(1)1443 408 y Fv(+)22 b Fl(\013)1603 423 y Fo(2)1643 327 y Fj(\000)1688 408 y Fl(n)h Fv(+)f Fl(p)p Fv(\()p Fl(\033)t Fv(\))2051 327 y Fj(\001)2118 408 y Fv(+)g Fl(\013)2278 423 y Fo(3)2400 340 y Fv(2)p Fl(t)p Fv(\()p Fl(\033)t Fv(\))p 2328 385 364 4 v 2328 476 a Fl(n)g Fu(\000)h Fl(p)p Fv(\()p Fl(\033)t Fv(\))2729 408 y Fu(\025)28 b Fv(0)p Fl(:)330 724 y Fv(W)-8 b(e)36 b(no)m(w)h(need)g(to)f(\014nd)g(out)g(whic)m(h)h(of)f(the)g(p)s (ossible)f(c)m(hoices)i(of)f Fl(p)p Fv(\()p Fl(\033)t Fv(\))p Fl(;)17 b(t)p Fv(\()p Fl(\033)t Fv(\))35 b(yield)h(equations) 330 941 y(of)41 b(facets)h(\(faces)g(of)g(dimension)e(2\),)j(and)f (whic)m(h)g(are)f(redundan)m(t)i(and)f(can)g(b)s(e)f(eliminated.)330 1157 y(Equiv)-5 b(alen)m(tly)d(,)40 b(w)m(e)g(can)g(adopt)f(the)g(dual) f(p)s(ersp)s(ectiv)m(e,)43 b(and)c(\014nd)h(the)f(extreme)h(ra)m(ys)g (of)e(the)330 1374 y(dual)h(cone.)65 b(This)39 b(is)g(a)g(tec)m(hnical) h(computation,)f(whic)m(h)h(w)m(e)h(ha)m(v)m(e)g(left)d(for)h(the)h (end)g(of)f(the)330 1591 y(section)33 b(\(Lemma)e(3.3.3\).)471 1807 y(Using)h(this)h(result,)f(in)g(the)h(case)g(of)g(ev)m(en)h Fl(n)p Fv(,)f(the)g(system)g(reduces)i(to)1617 2086 y Fl(\013)1679 2101 y Fo(1)1740 2086 y Fv(+)22 b Fl(n\013)1958 2101 y Fo(2)2026 2086 y Fu(\025)28 b Fv(0)1617 2303 y Fl(\013)1679 2318 y Fo(1)1740 2303 y Fv(+)22 b(\(2)p Fl(n)g Fu(\000)h Fv(3\))p Fl(\013)2254 2318 y Fo(2)2321 2303 y Fu(\025)28 b Fv(0)1617 2520 y Fl(\013)1679 2535 y Fo(1)1740 2520 y Fv(+)22 b(\(2)p Fl(n)g Fu(\000)h Fv(2\))p Fl(\013)2254 2535 y Fo(2)2315 2520 y Fv(+)f Fl(\013)2475 2535 y Fo(3)2542 2520 y Fu(\025)29 b Fv(0)1617 2736 y Fl(\013)1679 2751 y Fo(1)1740 2736 y Fv(+)22 b Fl(n\013)1958 2751 y Fo(2)2020 2736 y Fv(+)g Fl(\013)2180 2751 y Fo(3)2247 2736 y Fu(\025)28 b Fv(0)3699 2393 y(\(3.6\))330 2979 y(whereas)34 b(for)e(o)s(dd)g Fl(n)p Fv(,)h(the)g(system)h(is)e(equiv) -5 b(alen)m(t)32 b(to)1549 3258 y Fl(\013)1611 3273 y Fo(1)1673 3258 y Fv(+)22 b Fl(n\013)1891 3273 y Fo(2)1959 3258 y Fu(\025)28 b Fv(0)1549 3474 y Fl(\013)1611 3489 y Fo(1)1673 3474 y Fv(+)22 b(\(2)p Fl(n)g Fu(\000)h Fv(3\))p Fl(\013)2187 3489 y Fo(2)2254 3474 y Fu(\025)28 b Fv(0)1549 3691 y Fl(\013)1611 3706 y Fo(1)1673 3691 y Fv(+)22 b(\(2)p Fl(n)g Fu(\000)h Fv(2\))p Fl(\013)2187 3706 y Fo(2)2248 3691 y Fv(+)f Fl(\013)2408 3706 y Fo(3)2475 3691 y Fu(\025)28 b Fv(0)1549 3908 y Fl(\013)1611 3923 y Fo(1)1673 3908 y Fv(+)22 b(\()p Fl(n)g Fv(+)g(1\))p Fl(\013)2136 3923 y Fo(2)2198 3908 y Fv(+)g Fl(\013)2358 3923 y Fo(3)2425 3908 y Fu(\025)28 b Fv(0)1549 4124 y Fl(n\013)1669 4139 y Fo(1)1731 4124 y Fv(+)22 b Fl(n)1887 4088 y Fo(2)1927 4124 y Fl(\013)1989 4139 y Fo(2)2051 4124 y Fv(+)g(\()p Fl(n)g Fu(\000)h Fv(3\))p Fl(\013)2516 4139 y Fo(3)2582 4124 y Fu(\025)29 b Fv(0)p Fl(:)3699 3672 y Fv(\(3.7\))330 4372 y(The)39 b(set)g(of)f(all)e(feasible)i(3-tuples)f(\()p Fl(\013)1777 4387 y Fo(1)1817 4372 y Fl(;)17 b(\013)1923 4387 y Fo(2)1962 4372 y Fl(;)g(\013)2068 4387 y Fo(3)2107 4372 y Fv(\))38 b(is)g(a)g(p)s(olyhedral)f(cone,)k(whic)m(h,)f(for)e (ev)m(en)i Fl(n)p Fv(,)330 4589 y(has)d(at)f(most)g(4)g(extreme)h(ra)m (ys)h(and)e(for)g(o)s(dd)g Fl(n)h Fv(has)g(at)f(most)g(5)g(extreme)h (ra)m(ys.)56 b(W)-8 b(e)37 b(call)e(an)330 4805 y(inequalit)m(y)d(of)g (\(3.6\){\(3.7\))f Fm(active)h Fv(on)h(a)f(particular)f(tuple)h(if)f (it)h(holds)g(with)g(equalit)m(y)-8 b(.)471 5022 y(It)31 b(is)f(readily)f(v)m(eri\014ed)i(that)f(for)g(ev)m(en)i Fl(n)f Fv(the)g(follo)m(wing)c(tuples)k(span)g(the)g(extreme)g(ra)m(ys) g(of)p eop %%Page: 55 63 55 62 bop 2070 -210 a Fw(55)330 91 y Fv(the)33 b(set)g(of)f(solutions)g (to)g(\(3.6\):)942 295 y Fj(\000)987 376 y Fu(\000)p Fl(n;)115 b Fv(1)p Fl(;)f Fv(0)1503 295 y Fj(\001)1936 376 y Fv(4th)33 b(and)g(1st)f(inequalities)f(are)h(activ)m(e)985 512 y Fj(\000)1031 593 y Fv(0)p Fl(;)114 b Fv(0)p Fl(;)g Fv(1)1460 512 y Fj(\001)1936 593 y Fv(1st)33 b(and)g(2nd)f (inequalities)f(are)i(activ)m(e)832 729 y Fj(\000)878 809 y Fv(2)p Fl(n)22 b Fu(\000)g Fv(3)p Fl(;)114 b Fu(\000)p Fv(1)p Fl(;)h Fv(1)1613 729 y Fj(\001)1936 809 y Fv(2nd)33 b(and)g(3d)f(inequalities)f(are)i(activ)m(e)946 945 y Fj(\000)992 1026 y Fv(1)p Fl(;)114 b Fv(0)p Fl(;)g Fu(\000)p Fv(1)1498 945 y Fj(\001)1936 1026 y Fv(3d)33 b(and)g(4th)f (inequalities)f(are)h(activ)m(e)330 1268 y(and)f(that)g(for)g(o)s(dd)g Fl(n)g Fv(the)h(follo)m(wing)c(tuples)k(span)f(the)h(extreme)g(ra)m(ys) g(of)f(the)g(set)h(of)f(solutions)330 1485 y(to)h(\(3.7\):)942 1689 y Fj(\000)987 1769 y Fu(\000)p Fl(n;)115 b Fv(1)p Fl(;)f Fv(0)1503 1689 y Fj(\001)1980 1769 y Fv(5th)32 b(and)h(1st)g(inequalities)d(are)j(activ)m(e)985 1905 y Fj(\000)1031 1986 y Fv(0)p Fl(;)114 b Fv(0)p Fl(;)g Fv(1)1460 1905 y Fj(\001)1980 1986 y Fv(1st)33 b(and)f(2nd)h (inequalities)e(are)h(activ)m(e)832 2122 y Fj(\000)878 2203 y Fv(2)p Fl(n)22 b Fu(\000)g Fv(3)p Fl(;)114 b Fu(\000)p Fv(1)p Fl(;)h Fv(1)1613 2122 y Fj(\001)1980 2203 y Fv(2nd)33 b(and)f(3d)h(inequalities)d(are)j(activ)m(e)946 2339 y Fj(\000)992 2419 y Fv(1)p Fl(;)114 b Fv(0)p Fl(;)g Fu(\000)p Fv(1)1498 2339 y Fj(\001)1980 2419 y Fv(3d)32 b(and)h(4th)f(inequalities)f(are)i(activ)m(e)788 2555 y Fj(\000)834 2636 y Fu(\000)p Fv(2)p Fl(n)23 b Fu(\000)f Fv(3)p Fl(;)114 b Fv(3)p Fl(;)g Fu(\000)p Fl(n)1655 2555 y Fj(\001)1980 2636 y Fv(4th)32 b(and)h(5th)f(inequalities)f(are)i (activ)m(e)330 2878 y(W)-8 b(e)22 b(obtain)f Fl(r)824 2893 y Fo(1)863 2878 y Fl(;)c(r)951 2893 y Fo(2)991 2878 y Fl(;)g(r)1079 2893 y Fo(3)1118 2878 y Fl(;)g(r)1206 2893 y Fo(4)1267 2878 y Fv(and)22 b Fl(r)1490 2893 y Fo(5)p Fp(o)1585 2878 y Fv(b)m(y)h(scaling)d(the)j(corresp)s(onding)f (linear)e(com)m(binations)g Fl(\013)3698 2893 y Fo(1)3738 2878 y Fl(g)3785 2893 y Fo(1)3824 2878 y Fv(+)330 3095 y Fl(\013)392 3110 y Fo(2)431 3095 y Fl(g)478 3110 y Fo(2)537 3095 y Fv(+)f Fl(\013)694 3110 y Fo(3)733 3095 y Fl(g)780 3110 y Fo(3)851 3095 y Fv(so)31 b(that)g(the)g(v)-5 b(alue)31 b(at)g(the)g(iden)m(tit)m(y)g(is)g(equal)g(to)f(1)h(and)h (hence)g Fl(r)3291 3110 y Fo(1)3330 3095 y Fl(;)17 b(r)3418 3110 y Fo(2)3458 3095 y Fl(;)g(r)3546 3110 y Fo(3)3585 3095 y Fl(;)g(r)3673 3110 y Fo(4)3743 3095 y Fv(and)330 3311 y Fl(r)374 3326 y Fo(5)p Fp(o)480 3311 y Fv(lie)31 b(on)i(the)g(same)f(plane)g(in)g(span)q Fu(f)p Fl(g)1825 3326 y Fo(1)1864 3311 y Fl(;)17 b(g)1955 3326 y Fo(2)1993 3311 y Fl(;)g(g)2084 3326 y Fo(3)2123 3311 y Fu(g)p Fv(.)p 3829 3311 4 66 v 3833 3249 59 4 v 3833 3311 V 3891 3311 4 66 v 330 3603 a Fm(R)-5 b(emark)34 b(3.3.2)g(\(Gener)-5 b(ators)35 b(of)g(the)f(Centr)-5 b(al)35 b(Cone\).)48 b Fv(W)-8 b(e)47 b(observ)m(e)h(that)f Fl(r)3254 3618 y Fo(1)3340 3603 y Fv(and)f Fl(r)3587 3618 y Fo(5)p Fp(o)3708 3603 y Fv(ha)m(v)m(e)330 3820 y(spik)m(e)23 b(distributions)d(\(in)h (particular,)h Fl(r)1776 3835 y Fo(1)1837 3820 y Fv(has)g(the)g (\\sharp)g(spik)m(e")h(distribution)d(of)h(Section)g(2.4.1\))330 4036 y(corresp)s(onding)j(to)g(the)h(cases)g(of)f Fl(\015)1609 4051 y Fo(1)1676 4036 y Fv(=)j(1)d(and)h Fl(\015)2085 4051 y Fo(5)2151 4036 y Fv(=)j(1)c(resp)s(ectiv)m(ely)h(in)f(Theorem)g (2.4.1,)i(that)e Fl(r)3861 4051 y Fo(2)330 4253 y Fv(has)g(the)h(damp)s (ed)f(oscillator)d(distribution)h(corresp)s(onding)i(to)g(the)g(case)h (of)f Fl(\015)3190 4268 y Fo(2)3256 4253 y Fv(=)k(1)c(and)g(that)g Fl(r)3861 4268 y Fo(3)330 4470 y Fv(and)j Fl(r)558 4485 y Fo(4)624 4470 y Fv(ha)m(v)m(e)g(bullsey)m(e)h(distributions)d (corresp)s(onding)h(to)g(the)h(cases)h(of)e Fl(\015)3060 4485 y Fo(3)3127 4470 y Fv(=)i(1)e(and)h Fl(\015)3541 4485 y Fo(4)3607 4470 y Fv(=)h(1)e(re-)330 4686 y(sp)s(ectiv)m(ely)-8 b(.)43 b(If)28 b Fl(n)g Fv(is)g(ev)m(en)i(then)e Fl(r)1561 4701 y Fo(5)p Fp(o)1674 4686 y Fl(=)-60 b Fu(2)28 b Fl(K)7 b Fv(,)29 b(for)e(if)g Fl(\033)33 b Fv(is)27 b(a)h(pro)s(duct)h(of)e Fl(n=)p Fv(2)h(comm)m(uting)e(2-cycles,)330 4903 y(so)e(that)g Fl(p)p Fv(\()p Fl(\033)t Fv(\))j(=)h(0)c(and)g Fl(t)p Fv(\()p Fl(\033)t Fv(\))j(=)h Fl(n=)p Fv(2,)d(then)g Fl(r)1980 4918 y Fo(5)p Fp(o)2054 4903 y Fv(\()p Fl(\033)t Fv(\))i(=)h(\()p Fl(n)2416 4867 y Fo(2)2460 4903 y Fv(+)5 b Fl(n)g Fu(\000)g Fv(3\))p Fl(=)p Fv(\()p Fl(n)2918 4867 y Fo(2)2960 4903 y Fu(\000)g Fv(2)p Fl(n)g Fu(\000)g Fv(3\))27 b Fl(>)h Fv(1)f(=)h Fl(r)3678 4918 y Fo(5)p Fp(o)3751 4903 y Fv(\()p Fl(")p Fv(\).)p eop %%Page: 56 64 56 63 bop 2070 -210 a Fw(56)330 91 y Fs(3.3.1)105 b(Asymptotic)35 b(Geometry)f(of)h(the)g(Cen)m(tral)f(Cone)471 372 y Fv(W)-8 b(e)28 b(mak)m(e)g(some)g(brief)f(observ)-5 b(ations)28 b(on)g(the)g(structure)h(of)f(the)g(cen)m(tral)g(cone.)42 b(The)29 b(geom-)330 589 y(etry)40 b(of)g(the)g(cone)g(dep)s(ends)h(on) f(the)g(basis)g(c)m(hosen)h(for)e Fi(R)2553 553 y Fo(3)2598 589 y Fv(.)65 b(A)40 b(natural)e(c)m(hoice)i(for)g(a)f(basis)330 806 y(is)d(the)h(three)g(c)m(haracters)h Fl(\037)1386 821 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1578 806 y Fl(;)17 b(\037)1683 821 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(2)1911 806 y Fv(and)37 b Fl(\037)2166 821 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)2413 806 y Fv(.)55 b(The)38 b(co)s(ordinates)e(w)m(e)h(found)g(in)e(the)330 1022 y(pro)s(of)45 b(of)g(Lemma)f(3.3.1)h(are)h(with)f(resp)s(ect)i(to) e(the)h(basis)g Fl(g)2700 1037 y Fo(1)2739 1022 y Fl(;)17 b(g)2830 1037 y Fo(2)2869 1022 y Fl(;)g(g)2960 1037 y Fo(3)2999 1022 y Fv(.)83 b(Changing)45 b(basis)g(to)330 1239 y Fl(\037)391 1254 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)583 1239 y Fl(;)17 b(\037)688 1254 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(2)880 1239 y Fl(;)g(\037)985 1254 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)1264 1239 y Fv(w)m(e)34 b(get)e(the)h(follo)m(wing)d(generators:)1116 1494 y Fl(r)1160 1509 y Fo(1)1227 1494 y Fv(:)1454 1427 y(1)p 1364 1471 229 4 v 1364 1563 a Fl(n)22 b Fu(\000)h Fv(2)1603 1414 y Fj(\000)1648 1494 y Fu(\000)p Fl(n)g Fv(+)f(3)p Fl(;)114 b Fv(1)p Fl(;)g Fv(1)2333 1414 y Fj(\001)1116 1736 y Fl(r)1160 1751 y Fo(2)1227 1736 y Fv(:)1354 1656 y Fj(\000)1400 1736 y Fv(0)p Fl(;)g Fu(\000)p Fv(1)p Fl(;)g Fv(1)1906 1656 y Fj(\001)1116 1990 y Fl(r)1160 2005 y Fo(3)1227 1990 y Fv(:)1587 1923 y(1)p 1364 1967 496 4 v 1364 2059 a Fl(n)1422 2030 y Fo(2)1484 2059 y Fu(\000)22 b Fv(5)p Fl(n)g Fv(+)g(6)1869 1910 y Fj(\000)1915 1990 y Fv(2)p Fl(n)g Fu(\000)h Fv(6)p Fl(;)114 b Fu(\000)p Fv(2)p Fl(;)g Fv(0)2650 1910 y Fj(\001)1116 2256 y Fl(r)1160 2271 y Fo(4)1227 2256 y Fv(:)1454 2188 y(1)p 1364 2233 229 4 v 1364 2324 a Fl(n)22 b Fu(\000)h Fv(2)1603 2175 y Fj(\000)1648 2256 y Fv(1)p Fl(;)114 b Fv(1)p Fl(;)g Fu(\000)p Fv(1)2154 2175 y Fj(\001)1099 2521 y Fl(r)1143 2536 y Fo(5)p Fp(o)1244 2521 y Fv(:)1588 2454 y(1)p 1364 2499 497 4 v 1364 2590 a Fl(n)1422 2561 y Fo(2)1484 2590 y Fu(\000)22 b Fv(2)p Fl(n)g Fu(\000)h Fv(3)1871 2441 y Fj(\000)1916 2521 y Fu(\000)p Fv(2)p Fl(n)g Fu(\000)g Fv(6)p Fl(;)114 b(n)22 b Fv(+)g(3)p Fl(;)114 b Fu(\000)p Fl(n)23 b Fv(+)f(3)3086 2441 y Fj(\001)330 2743 y Fv(F)-8 b(rom)24 b(this)g(data)h(w)m(e)h(can)f(compute)g(the)h(angles)f(b)s(et) m(w)m(een)i(the)e(generators)h(with)e(resp)s(ect)j(to)d(the)330 2960 y(basis)39 b Fl(\037)637 2975 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)829 2960 y Fl(;)17 b(\037)934 2975 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(2)1126 2960 y Fl(;)g(\037)1231 2975 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)1478 2960 y Fv(.)63 b(Tw)m(o)40 b(facts)f(are)g(w)m(orth)h(noting.)62 b(First,)40 b(the)g(angle)e(b)s(et)m(w)m(een)330 3176 y(the)32 b(\(opp)s(osite\))f(extreme)h(ra)m(ys)h Fl(r)1580 3191 y Fo(1)1651 3176 y Fv(and)e Fl(r)1883 3191 y Fo(3)1954 3176 y Fv(is)g(asymptotically)f Fl(\031)t Fv(,)h(so)h(the)g(cone)g(b)s (ecomes)g(v)m(ery)330 3393 y(wide.)43 b(Second,)31 b(the)f(function)g Fl(r)1539 3408 y Fo(5)p Fp(o)1642 3393 y Fv(approac)m(hes)h(the)f (plane)g(de\014ned)h(b)m(y)g Fl(r)3074 3408 y Fo(1)3143 3393 y Fv(and)f Fl(r)3374 3408 y Fo(4)3443 3393 y Fv(as)g Fl(n)e Fu(!)f(1)p Fv(.)330 3610 y(In)41 b(fact)f(the)h(distance)g(b)s (et)m(w)m(een)i(the)e(p)s(oin)m(t)f(of)g Fl(r)2211 3625 y Fo(5)p Fp(o)2325 3610 y Fv(on)g(the)h(unit)f(sphere)i(and)f(this)f (plane)g(is)330 3826 y Fl(O)s Fv(\()p Fl(n)504 3790 y Ff(\000)p Fo(2)598 3826 y Fv(\).)j(So)33 b Fl(r)886 3841 y Fo(5)p Fp(o)992 3826 y Fv(is)f(\\almost")e(a)i(linear)f(com)m (bination)g(of)h Fl(r)2565 3841 y Fo(1)2637 3826 y Fv(and)h Fl(r)2871 3841 y Fo(4)2943 3826 y Fv(for)f(large)f Fl(n)p Fv(.)471 4043 y(W)-8 b(e)29 b(conclude)h(this)e(section)h(with)g(the)g (pro)s(of)f(of)h(tec)m(hnical)f(lemma)f(used)j(in)e(Lemma)f(3.3.1.)330 4311 y Ft(Lemma)37 b(3.3.3.)147 b Fd(F)-8 b(or)32 b(a)g(p)s(erm)m (utation)f Fl(\033)h Fu(2)c Fl(S)2190 4326 y Fp(n)2237 4311 y Fd(,)33 b Fl(\033)e Fu(6)p Fv(=)d Fl(")p Fd(,)k(let)g Fl(a)2784 4326 y Fp(\033)2859 4311 y Fu(2)c Fi(R)3019 4275 y Fo(2)3097 4311 y Fd(b)s(e)33 b(the)g(p)s(oin)m(t)1573 4595 y Fl(a)1624 4610 y Fp(\033)1699 4595 y Fv(=)1802 4485 y Fj(\020)1862 4595 y Fl(p)p Fv(\()p Fl(\033)t Fv(\))p Fl(;)2269 4528 y Fv(2)p Fl(t)p Fv(\()p Fl(\033)t Fv(\))p 2197 4572 364 4 v 2197 4664 a Fl(n)22 b Fu(\000)h Fl(p)p Fv(\()p Fl(\033)t Fv(\))2571 4485 y Fj(\021)2630 4595 y Fl(:)330 4879 y Fd(Let)33 b Fl(P)41 b Fv(=)27 b(con)m(v)909 4798 y Fj(\010)967 4879 y Fl(a)1018 4894 y Fp(\033)1093 4879 y Fv(:)g Fl(\033)32 b Fu(6)p Fv(=)c Fl(")1384 4798 y Fj(\011)1474 4879 y Fd(b)s(e)33 b(the)g(con)m(v)m(ex)i(h)m(ull)c(of)h (all)f(suc)m(h)j(p)s(oin)m(ts)e Fl(a)3098 4894 y Fp(\033)3145 4879 y Fd(.)471 5096 y(If)h Fl(n)f Fd(is)g(ev)m(en,)j(the)e(extreme)g (p)s(oin)m(ts)f(of)g Fl(P)46 b Fd(are)1170 5380 y Fv(\(0)p Fl(;)17 b Fv(0\))p Fl(;)113 b Fv(\()p Fl(n)22 b Fu(\000)h Fv(3)p Fl(;)17 b Fv(0\))p Fl(;)114 b Fv(\()p Fl(n)22 b Fu(\000)g Fv(2)p Fl(;)17 b Fv(1\))97 b Fd(and)h Fv(\(0)p Fl(;)17 b Fv(1\))p Fl(:)p eop %%Page: 57 65 57 64 bop 2070 -210 a Fw(57)471 91 y Fd(If)33 b Fl(n)f Fd(is)g(o)s(dd,)h(the)g(extreme)g(p)s(oin)m(ts)f(of)h Fl(P)45 b Fd(are)753 408 y Fv(\(0)p Fl(;)17 b Fv(0\))p Fl(;)113 b Fv(\()p Fl(n)22 b Fu(\000)h Fv(3)p Fl(;)17 b Fv(0\))p Fl(;)113 b Fv(\()p Fl(n)23 b Fu(\000)f Fv(2)p Fl(;)17 b Fv(1\))p Fl(;)2188 327 y Fj(\000)2233 408 y Fv(0)p Fl(;)g Fv(\()p Fl(n)22 b Fu(\000)h Fv(3\))p Fl(=n)2738 327 y Fj(\001)2881 408 y Fd(and)195 b Fv(\(1)p Fl(;)17 b Fv(1\))p Fl(:)330 724 y Fm(Pr)-5 b(o)g(of.)45 b Fv(The)31 b(set)f(of)f(all)e(p)s(ossible)i(v)-5 b(alues)1871 643 y Fj(\000)1916 724 y Fl(p)p Fv(\()p Fl(\033)t Fv(\))p Fl(;)17 b(t)p Fv(\()p Fl(\033)t Fv(\))2314 643 y Fj(\001)2360 724 y Fv(,)30 b(where)h Fl(\033)g Fu(6)p Fv(=)d Fl(")p Fv(,)i(consists)g(of)f(all)e(pairs)i(of)330 941 y(non-negativ)m(e)23 b(in)m(tegers)g(\()p Fl(p;)17 b(t)p Fv(\))23 b(suc)m(h)h(that)f Fl(p)k Fu(\024)h Fl(n)r Fu(\000)r Fv(2,)e(2)p Fl(t)i Fu(\024)g Fl(n)23 b Fv(and,)i(additionally)-8 b(,)22 b Fl(p)r Fv(+)r(2)p Fl(t)28 b Fu(\024)g Fl(n)r Fu(\000)r Fv(3)330 1157 y(or)h Fl(p)16 b Fv(+)g(2)p Fl(t)28 b Fv(=)f Fl(n)p Fv(.)43 b(T)-8 b(o)29 b(\014nd)h(the)g(extreme)g(p)s(oin)m(ts)g (of)f(the)h(set)g(of)f(feasible)g(p)s(oin)m(ts)3264 1077 y Fj(\000)3309 1157 y Fl(p;)17 b Fv(2)p Fl(t=)p Fv(\()p Fl(n)f Fu(\000)g Fl(p)p Fv(\))3827 1077 y Fj(\001)3873 1157 y Fv(,)330 1374 y(w)m(e)40 b(c)m(ho)s(ose)h(a)e(generic)g(v)m (ector)h(\()p Fl(\015)1610 1389 y Fo(1)1649 1374 y Fl(;)17 b(\015)1744 1389 y Fo(2)1783 1374 y Fv(\))39 b(and)g(in)m(v)m(estigate) h(for)f(whic)m(h)g(v)-5 b(alues)40 b(of)e Fl(p)i Fv(and)f Fl(t)g Fv(the)330 1591 y(maxim)m(um)30 b(of)1816 1807 y Fl(\015)1867 1822 y Fo(1)1906 1807 y Fl(p)22 b Fv(+)g Fl(\015)2126 1822 y Fo(2)2248 1740 y Fv(2)p Fl(t)p 2175 1784 229 4 v 2175 1876 a(n)g Fu(\000)h Fl(p)330 2078 y Fv(is)32 b(attained.)471 2295 y(Clearly)-8 b(,)29 b(w)m(e)i(can)f (assume)g(that)g Fl(\015)1737 2310 y Fo(2)1803 2295 y Fu(6)p Fv(=)e(0.)42 b(If)30 b Fl(\015)2171 2310 y Fo(2)2237 2295 y Fl(<)e Fv(0)h(then)i(w)m(e)f(should)g(c)m(ho)s(ose)g(the)g (smallest)330 2511 y(p)s(ossible)39 b Fl(t)g Fv(whic)m(h)h(w)m(ould)f (b)s(e)h Fl(t)g Fv(=)f(0)g(unless)h Fl(p)f Fv(=)g Fl(n)27 b Fu(\000)g Fv(2)40 b(when)g(w)m(e)g(ha)m(v)m(e)h(to)e(c)m(ho)s(ose)h Fl(t)g Fv(=)f(1.)330 2728 y(Dep)s(ending)32 b(on)h(the)g(sign)f(of)g Fl(\015)1486 2743 y Fo(1)1525 2728 y Fv(,)g(this)h(pro)s(duces)g(the)g (follo)m(wing)d(pairs)1221 3044 y(\()p Fl(p;)17 b(t)p Fv(\))27 b(=)1556 2934 y Fj(n)1622 3044 y Fv(\(0)p Fl(;)17 b Fv(0\))p Fl(;)113 b Fv(\()p Fl(n)23 b Fu(\000)f Fv(3)p Fl(;)17 b Fv(0\))p Fl(;)114 b Fv(\()p Fl(n)22 b Fu(\000)h Fv(2)p Fl(;)17 b Fv(1\))2917 2934 y Fj(o)2982 3044 y Fl(:)330 3361 y Fv(If)36 b Fl(\015)482 3376 y Fo(2)554 3361 y Fl(>)d Fv(0)j(then)g(the)h(largest)e(p)s(ossible)g(v)-5 b(alue)35 b(of)h(2)p Fl(t=)p Fv(\()p Fl(n)24 b Fu(\000)h Fl(p)p Fv(\))36 b(is)f(1.)53 b(If)36 b Fl(\015)3063 3376 y Fo(1)3135 3361 y Fl(>)e Fv(0)h(this)h(pro)s(duces)330 3577 y(the)d(\(already)f(included\))g(p)s(oin)m(t)1732 3794 y(\()p Fl(p;)17 b(t)p Fv(\))27 b(=)g(\()p Fl(n)c Fu(\000)f Fv(2)p Fl(;)17 b Fv(1)2426 3713 y Fj(\001)2471 3794 y Fl(:)330 4065 y Fv(If)33 b Fl(\015)479 4080 y Fo(1)545 4065 y Fl(<)28 b Fv(0)k(w)m(e)i(get)1522 4281 y(\()p Fl(p;)17 b(t)p Fv(\))27 b(=)h(\(0)p Fl(;)17 b(n=)p Fv(2\))97 b(for)32 b(ev)m(en)i Fl(n)330 4552 y Fv(and)979 4769 y(\()p Fl(p;)17 b(t)p Fv(\))27 b(=)1314 4658 y Fj(n)1380 4688 y(\000)1426 4769 y Fv(0)p Fl(;)17 b Fv(\()p Fl(n)k Fu(\000)i Fv(3\))p Fl(=)p Fv(2)1921 4688 y Fj(\001)1966 4769 y Fl(;)2010 4688 y Fj(\000)2055 4769 y Fv(1)p Fl(;)17 b Fv(\()p Fl(n)22 b Fu(\000)h Fv(1\))p Fl(=)p Fv(2)2551 4688 y Fj(\001)2596 4658 y(o)2760 4769 y Fv(for)32 b(o)s(dd)97 b Fl(n:)330 5039 y Fv(Summarizing,)30 b(the)j(extreme)g(p)s(oin)m(ts)f (of)g Fl(P)46 b Fv(are)977 5356 y(\(0)p Fl(;)17 b Fv(0\))p Fl(;)113 b Fv(\()p Fl(n)22 b Fu(\000)h Fv(3)p Fl(;)17 b Fv(0\))p Fl(;)113 b Fv(\()p Fl(n)23 b Fu(\000)f Fv(2)p Fl(;)17 b Fv(1\))p Fl(;)114 b Fv(\(0)p Fl(;)17 b Fv(1\))194 b(for)32 b(ev)m(en)i Fl(n)p eop %%Page: 58 66 58 65 bop 2070 -210 a Fw(58)330 91 y Fv(and)623 408 y(\(0)p Fl(;)17 b Fv(0\))p Fl(;)114 b Fv(\()p Fl(n)22 b Fu(\000)g Fv(3)p Fl(;)17 b Fv(0\))p Fl(;)114 b Fv(\()p Fl(n)22 b Fu(\000)h Fv(2)p Fl(;)17 b Fv(1\))p Fl(;)2058 327 y Fj(\000)2104 408 y Fv(0)p Fl(;)g Fv(\()p Fl(n)22 b Fu(\000)g Fv(3\))p Fl(=n)2608 327 y Fj(\001)2654 408 y Fl(;)114 b Fv(\(1)p Fl(;)17 b Fv(1\))194 b(for)32 b(o)s(dd)h Fl(n)330 724 y Fv(as)g(claimed.)p 3829 724 4 66 v 3833 661 59 4 v 3833 724 V 3891 724 4 66 v 330 990 a Ft(3.4)112 b(Some)37 b(Estimates)471 1306 y Fv(In)31 b(this)f(section)h(w)m(e)g(mak)m(e)g (some)f(estimates)g(of)g(the)h(n)m(um)m(b)s(er)g(of)f(p)s(erm)m (utations)g(satisfying)330 1522 y(certain)e(conditions.)41 b(W)-8 b(e)28 b(need)h(these)h(estimates)d(to)h(pro)m(v)m(e)h(the)g (distributional)c(results)j(in)f(the)330 1739 y(follo)m(wing)j (sections.)471 1956 y(W)-8 b(e)38 b(b)s(egin)f(with)g(a)h(Mark)m(o)m(v) g(t)m(yp)s(e)h(estimate,)f(whic)m(h)g(asserts,)j(roughly)-8 b(,)38 b(that)f(a)g(function)330 2172 y(with)32 b(a)g(su\016cien)m(tly) i(large)d(a)m(v)m(erage)j(tak)m(es)g(su\016cien)m(tly)f(large)e(v)-5 b(alues)33 b(su\016cien)m(tly)g(often.)330 2464 y Ft(Lemma)k(3.4.1.)147 b Fd(Let)32 b Fl(X)39 b Fd(b)s(e)32 b(a)f(\014nite)g(set)h(and)g(let)f Fl(f)38 b Fv(:)28 b Fl(X)36 b Fu(\000)-17 b(!)28 b Fi(R)42 b Fd(b)s(e)32 b(a)f(function.)43 b(Supp)s(ose)330 2681 y(that)32 b Fl(f)11 b Fv(\()p Fl(x)p Fv(\))28 b Fu(\024)g Fv(1)33 b Fd(for)f(all)e Fl(x)e Fu(2)g Fl(X)41 b Fd(and)32 b(that)1395 2929 y Fv(1)p 1348 2974 145 4 v 1348 3065 a Fu(j)p Fl(X)8 b Fu(j)1521 2902 y Fj(X)1518 3114 y Fp(x)p Ff(2)p Fp(X)1685 2997 y Fl(f)j Fv(\()p Fl(x)p Fv(\))28 b Fu(\025)g Fl(\014)103 b Fd(for)32 b(some)98 b Fl(\014)33 b(>)27 b Fv(0)p Fl(:)330 3313 y Fd(Then)34 b(for)e(an)m(y)h Fv(0)27 b Fl(<)h(\015)33 b(<)27 b Fv(1)33 b Fd(w)m(e)g(ha)m(v)m(e)1296 3545 y Fj(\014)1296 3605 y(\014)1329 3549 y(\010)1387 3629 y Fl(x)c Fu(2)f Fl(X)35 b Fv(:)28 b Fl(f)11 b Fv(\()p Fl(x)p Fv(\))28 b Fu(\025)g Fl(\014)6 b(\015)2176 3549 y Fj(\011)2233 3545 y(\014)2233 3605 y(\014)2294 3629 y Fu(\025)29 b Fl(\014)6 b Fv(\(1)21 b Fu(\000)i Fl(\015)5 b Fv(\))p Fu(j)p Fl(X)j Fu(j)p Fl(:)330 3946 y Fm(Pr)-5 b(o)g(of.)45 b Fv(W)-8 b(e)33 b(ha)m(v)m(e)945 4225 y Fl(\014)g Fu(\024)1196 4157 y Fv(1)p 1148 4202 V 1148 4293 a Fu(j)p Fl(X)8 b Fu(j)1322 4130 y Fj(X)1319 4342 y Fp(x)p Ff(2)p Fp(X)1486 4225 y Fl(f)j Fv(\()p Fl(x)p Fv(\))27 b(=)1864 4157 y(1)p 1817 4202 V 1817 4293 a Fu(j)p Fl(X)8 b Fu(j)2082 4130 y Fj(X)1987 4346 y Fp(x)p Fo(:)p Fp(f)f Fo(\()p Fp(x)p Fo(\))p Fp(<\014)s(\015)2337 4225 y Fl(f)k Fv(\()p Fl(x)p Fv(\))23 b(+)2705 4157 y(1)p 2657 4202 V 2657 4293 a Fu(j)p Fl(X)8 b Fu(j)2923 4130 y Fj(X)2828 4346 y Fp(x)p Fo(:)p Fp(f)f Fo(\()p Fp(x)p Fo(\))p Ff(\025)p Fp(\014)s(\015)3178 4225 y Fl(f)k Fv(\()p Fl(x)p Fv(\))945 4562 y Fu(\024)28 b Fl(\014)6 b(\015)27 b Fv(+)1297 4403 y Fj(\014)1297 4463 y(\014)1330 4407 y(\010)1388 4488 y Fl(x)h Fv(:)g Fl(f)11 b Fv(\()p Fl(x)p Fv(\))28 b Fu(\025)g Fl(\014)6 b(\015)1966 4407 y Fj(\011)2023 4403 y(\014)2023 4463 y(\014)p 1297 4539 761 4 v 1605 4630 a Fu(j)p Fl(X)i Fu(j)2067 4562 y Fl(:)330 4831 y Fv(Hence)1410 4963 y Fj(\014)1410 5023 y(\014)1443 5047 y Fu(f)p Fl(x)28 b Fv(:)g Fl(f)11 b Fv(\()p Fl(x)p Fv(\))27 b Fu(\025)h Fl(\014)6 b(\015)f Fu(g)2120 4963 y Fj(\014)2120 5023 y(\014)2181 5047 y Fu(\025)28 b Fl(\014)6 b Fv(\(1)21 b Fu(\000)i Fl(\015)5 b Fv(\))p Fu(j)p Fl(X)j Fu(j)p Fl(:)p 3829 5318 4 66 v 3833 5256 59 4 v 3833 5318 V 3891 5318 4 66 v eop %%Page: 59 67 59 66 bop 2070 -210 a Fw(59)471 91 y Fv(W)-8 b(e)35 b(next)g(estimate)e (the)i(fraction)e(of)h(p)s(erm)m(utations)f(that)h(ha)m(v)m(e)h(no)f (\014xed)i(p)s(oin)m(ts)e(and)g(no)330 308 y(2-cycles.)330 586 y Ft(Lemma)j(3.4.2)h(\(P)m(erm)m(utations)d(with)h(no)i(Fixed)g(P)m (oin)m(ts)d(and)k(2-cycles\).)146 b Fd(The)22 b(frac-)330 802 y(tion)31 b(of)g(p)s(erm)m(utations)g Fl(\033)h Fu(2)c Fl(S)1477 817 y Fp(n)1556 802 y Fd(without)j(\014xed)i(p)s(oin)m(ts)f (is)f Fl(e)2588 766 y Ff(\000)p Fo(1)2683 802 y Fv(\(1)20 b(+)h Fl(o)p Fv(\(1\)\))p Fd(.)42 b(More)33 b(precisely)-8 b(,)32 b(it)330 1019 y(is)k(equal)g(to)g Fl(d)867 1034 y Fp(n)914 1019 y Fd(,)h(see)h(De\014nition)c(1.1.8.)55 b(Similarly)-8 b(,)33 b(the)k(fraction)e(of)h(p)s(erm)m(utations)f (without)330 1236 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ff 120.00 scf sf 1259 972 m gs 1 -1 sc (2) col-1 sh gr /Times-Roman ff 120.00 scf sf 5110 972 m gs 1 -1 sc (2) col-1 sh gr /Times-Roman ff 120.00 scf sf 1589 3282 m gs 1 -1 sc (6e) col-1 sh gr /Times-Roman ff 120.00 scf sf 5414 3869 m gs 1 -1 sc (6o) col-1 sh gr /Times-Roman ff 120.00 scf sf 5414 3282 m gs 1 -1 sc (6e) col-1 sh gr $F2psEnd rs %%EndDocument @endspecial 1378 4998 a Fw(Figure)i(3.2:)42 b(The)30 b(Cen)m(tral)g(\(Pure\))g(Cone)330 5380 y Fm(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(2.2.1.)44 b Fv(W)-8 b(e)36 b(pro)s(ceed)f(as)g(in)f(the)h(pro)s(of)f(of)h(Theorem)g(2.1.1)f (\(Section)g(3.5\))p eop %%Page: 65 73 65 72 bop 2070 -210 a Fw(65)330 91 y Fv(with)34 b(some)g(mo)s (di\014cations.)47 b(Without)34 b(loss)g(of)g(generalit)m(y)-8 b(,)35 b(w)m(e)g(assume)g(that)f(the)h(maxim)m(um)330 308 y(of)f Fl(f)491 323 y Fo(0)530 308 y Fv(\()p Fl(\033)t Fv(\))g(is)g(attained)g(on)g(the)h(iden)m(tit)m(y)f(p)s(erm)m(utation)e Fl(")i Fv(and)h(that)f Fl(f)2947 323 y Fo(0)2986 308 y 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Ff(\000)p Fo(2)p Fp(;)p Fo(2)3743 958 y Fv(and)330 1175 y Fl(\037)391 1190 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)673 1175 y Fv(alone.)52 b(Moreo)m(v)m(er,)37 b(b)m(y)g(P)m(art)f(\(3\))f(of)g(Lemma)f(3.1.4,)h (w)m(e)i(ha)m(v)m(e)g(1)32 b(=)g Fl(g)t Fv(\()p Fl(")p Fv(\))g Fu(\025)h Fl(g)t Fv(\()p Fl(\033)t Fv(\))h(for)330 1391 y(all)i Fl(\033)k Fu(2)d Fl(S)729 1406 y Fp(n)776 1391 y Fv(.)60 b(Hence)39 b Fl(g)i Fv(lies)c(in)g(the)h(cen)m(tral)g (cone)g Fl(K)2350 1406 y Fp(p)2390 1391 y Fv(.)59 b(F)-8 b(rom)37 b(our)g(description)h(of)f(the)h(cone)330 1608 y(ab)s(o)m(v)m(e,)29 b(w)m(e)g(conclude)f(that)f Fl(g)j Fv(m)m(ust)e(b)s(e)f(a)g(con)m(v)m(ex)j(com)m(bination)25 b(of)i Fl(r)2902 1623 y Fo(1)2969 1608 y Fv(and)g Fl(r)3197 1623 y Fo(2)p Fp(e)3296 1608 y Fv(for)g Fl(n)h Fv(ev)m(en)h(and)330 1825 y(a)g(con)m(v)m(ex)i(com)m(bination)c(of)h Fl(r)1427 1840 y Fo(1)1496 1825 y Fv(and)h Fl(r)1726 1840 y Fo(2)p Fp(o)1828 1825 y Fv(for)f Fl(n)i Fv(o)s(dd.)42 b(Applying)28 b(P)m(art)h(\(1\))g(of)f(Lemma)g(3.1.4,)h(w)m(e)330 2042 y(can)38 b(replace)f(the)h(a)m(v)m(erage)g(of)f Fl(f)1540 2057 y Fo(0)1617 2042 y Fv(o)m(v)m(er)h(the)g(set)g Fl(U)10 b Fv(\()p Fl(";)17 b(k)s Fv(\))37 b(b)m(y)h(the)g(a)m(v)m(erage)g(of)f Fl(g)k Fv(o)m(v)m(er)d Fl(U)10 b Fv(\()p Fl(";)17 b(k)s Fv(\).)330 2258 y(The)35 b(pro)s(of)f(no)m(w)h(follo)m(ws)e(b)m(y)j (Lemma)d(3.4.3)h(and)h(the)f(observ)-5 b(ation)34 b(that)h Fl(\033)g Fu(2)c Fl(U)10 b Fv(\()p Fl(";)17 b(k)s Fv(\))34 b(if)g(and)330 2475 y(only)e(if)f Fl(p)p Fv(\()p Fl(\033)t Fv(\))d(=)g Fl(k)s Fv(.)p 3829 2475 4 66 v 3833 2412 59 4 v 3833 2475 V 3891 2475 4 66 v 471 2762 a(T)-8 b(o)33 b(pro)m(v)m(e)h(Theorem)e(2.2.2,)h(w)m(e)g(need)h(need)f(one)g (preliminary)d(result.)330 3030 y Ft(Lemma)37 b(3.6.2.)147 b Fd(Let)25 b Fl(g)j Fd(b)s(e)d(a)f(linear)f(com)m(bination)g(of)h Fl(g)2536 3045 y Fo(1)2603 3030 y Fv(=)k Fl(\037)2768 3045 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(2)2966 3030 y Fv(+)6 b Fl(\037)3109 3045 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)3384 3030 y Fv(=)27 b Fl(p)3536 2994 y Fo(2)3582 3030 y Fu(\000)6 b Fv(3)p Fl(p)g Fv(+)g(1)330 3247 y Fd(and)29 b Fl(g)563 3262 y Fo(2)630 3247 y Fv(=)e Fl(\037)794 3262 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)1056 3247 y Fu(\000)15 b Fl(\037)1209 3262 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(2)1429 3247 y Fv(=)28 b(1)15 b Fu(\000)g Fv(2)p Fl(t)29 b Fd(suc)m(h)h(that)f Fl(g)t Fv(\()p Fl(")p Fv(\))d(=)i(1)p Fd(.)42 b(F)-8 b(or)28 b(a)h Fv(3)e Fu(\024)i Fl(k)h Fu(\024)e Fl(n)15 b Fu(\000)g Fv(1)p Fd(,)30 b(let)f Fl(\033)3728 3262 y Fp(k)3800 3247 y Fd(b)s(e)330 3464 y(a)34 b(p)s(erm)m(utation)f(suc)m (h)j(that)e Fl(p)p Fv(\()p Fl(\033)1550 3479 y Fp(k)1593 3464 y Fv(\))d(=)f Fl(k)38 b Fd(and)c Fl(t)p Fv(\()p Fl(\033)2176 3479 y Fp(k)2220 3464 y Fv(\))c(=)h(0)j Fd(and)h(let)e Fl(\022)2857 3479 y Fp(k)2935 3464 y Fd(b)s(e)h(a)g(p)s (erm)m(utation)f(suc)m(h)330 3680 y(that)f Fl(p)p Fv(\()p Fl(\022)673 3695 y Fp(k)716 3680 y Fv(\))c(=)g Fl(k)35 b Fd(and)e Fl(t)p Fv(\()p Fl(\022)1280 3695 y Fp(k)1323 3680 y Fv(\))28 b(=)f(1)p Fd(.)43 b(Then)1387 3964 y Fv(max)1569 3884 y Fj(\010)1627 3964 y Fl(g)t Fv(\()p Fl(\033)1771 3979 y Fp(k)1813 3964 y Fv(\))p Fl(;)17 b(g)t Fv(\()p Fl(\022)2029 3979 y Fp(k)2071 3964 y Fv(\))2109 3884 y Fj(\011)2195 3964 y Fu(\025)2314 3897 y Fl(k)2368 3861 y Fo(2)2430 3897 y Fu(\000)23 b Fv(3)p Fl(k)i Fv(+)d(1)p 2310 3942 496 4 v 2310 4033 a Fl(n)2368 4004 y Fo(2)2430 4033 y Fu(\000)h Fv(3)p Fl(n)f Fv(+)g(1)2816 3964 y Fl(:)330 4249 y Fm(Pr)-5 b(o)g(of.)45 b Fv(Since)33 b Fl(g)t Fv(\()p Fl(")p Fv(\))26 b(=)i(1,)k Fl(g)1345 4264 y Fo(1)1384 4249 y Fv(\()p Fl(")p Fv(\))c(=)f Fl(n)1695 4213 y Fo(2)1757 4249 y Fu(\000)c Fv(3)p Fl(n)f Fv(+)g(1)32 b(and)h Fl(g)2402 4264 y Fo(2)2441 4249 y Fv(\()p Fl(")p Fv(\))27 b(=)h(1,)k(w)m(e)i(can) e(write)1440 4533 y Fl(g)f Fv(=)c Fl(\013)1683 4548 y Fo(1)1742 4466 y Fl(p)1791 4429 y Fo(2)1853 4466 y Fu(\000)22 b Fv(3)p Fl(p)g Fv(+)g(1)p 1733 4510 V 1733 4601 a Fl(n)1791 4572 y Fo(2)1853 4601 y Fu(\000)g Fv(3)p Fl(n)g Fv(+)g(1)2260 4533 y(+)g Fl(\013)2420 4548 y Fo(2)2460 4533 y Fv(\(1)g Fu(\000)g Fv(2)p Fl(t)p Fv(\))330 4817 y(for)32 b(some)g Fl(\013)785 4832 y Fo(1)857 4817 y Fv(and)h Fl(\013)1109 4832 y Fo(2)1181 4817 y Fv(suc)m(h)h(that)f Fl(\013)1675 4832 y Fo(1)1736 4817 y Fv(+)22 b Fl(\013)1896 4832 y Fo(2)1963 4817 y Fv(=)28 b(1.)43 b(Then)1432 5060 y Fl(g)t Fv(\()p Fl(\033)1576 5075 y Fp(k)1619 5060 y Fv(\))27 b(=)h Fl(\013)1850 5075 y Fo(1)1903 4992 y Fl(k)1957 4956 y Fo(2)2019 4992 y Fu(\000)23 b Fv(3)p Fl(k)i Fv(+)d(1)p 1899 5037 V 1899 5128 a Fl(n)1957 5099 y Fo(2)2019 5128 y Fu(\000)h Fv(3)p Fl(n)f Fv(+)g(1)2427 5060 y(+)g Fl(\013)2587 5075 y Fo(2)2724 5060 y Fv(and)1432 5349 y Fl(g)t Fv(\()p Fl(\022)1566 5364 y Fp(k)1609 5349 y Fv(\))27 b(=)h Fl(\013)1840 5364 y Fo(1)1893 5281 y Fl(k)1947 5245 y Fo(2)2009 5281 y Fu(\000)23 b Fv(3)p Fl(k)i Fv(+)d(1)p 1889 5326 V 1889 5417 a Fl(n)1947 5388 y Fo(2)2009 5417 y Fu(\000)h Fv(3)p Fl(n)f Fv(+)g(1)2417 5349 y Fu(\000)g Fl(\013)2578 5364 y Fo(2)p eop %%Page: 66 74 66 73 bop 2070 -210 a Fw(66)330 91 y Fv(W)-8 b(e)33 b(observ)m(e)h (that)f Fl(g)t Fv(\()p Fl(\033)1201 106 y Fp(k)1243 91 y Fv(\))f(and)h Fl(g)t Fv(\()p Fl(\022)1637 106 y Fp(k)1680 91 y Fv(\))f(are)h(linear)e(functions)h(of)g Fl(\013)2777 106 y Fo(1)2849 91 y Fv(and)h Fl(\013)3101 106 y Fo(2)3173 91 y Fv(and)g(that)f(for)1657 367 y Fl(\013)1719 382 y Fo(1)1787 367 y Fv(=)27 b(1)97 b(and)h Fl(\013)2353 382 y Fo(2)2420 367 y Fv(=)28 b(0)330 642 y(w)m(e)34 b(ha)m(v)m(e)1507 918 y Fl(g)t Fv(\()p Fl(\033)1651 933 y Fp(k)1693 918 y Fv(\))28 b(=)f Fl(g)t Fv(\()p Fl(\022)1996 933 y Fp(k)2039 918 y Fv(\))g(=)2222 850 y Fl(k)2276 814 y Fo(2)2338 850 y Fu(\000)22 b Fv(3)p Fl(k)j Fv(+)d(1)p 2218 895 496 4 v 2218 986 a Fl(n)2276 957 y Fo(2)2338 986 y Fu(\000)g Fv(3)p Fl(n)g Fv(+)g(1)3699 918 y(\(3.8\))330 1193 y(Let)876 1410 y Fl(\025)933 1425 y Fo(1)1000 1410 y Fv(=)1172 1343 y Fl(n)1230 1306 y Fo(2)1292 1343 y Fu(\000)h Fv(3)p Fl(n)f Fv(+)g(1)p 1114 1387 612 4 v 1114 1478 a(2\()p Fl(k)1255 1450 y Fo(2)1316 1478 y Fu(\000)h Fv(3)p Fl(k)i Fv(+)d(1\))1758 1410 y(+)1866 1343 y(1)p 1866 1387 49 4 v 1866 1478 a(2)2022 1410 y(and)98 b Fl(\025)2334 1425 y Fo(2)2401 1410 y Fv(=)2572 1343 y Fl(n)2630 1306 y Fo(2)2692 1343 y Fu(\000)23 b Fv(3)p Fl(n)f Fv(+)g(1)p 2514 1387 612 4 v 2514 1478 a(2\()p Fl(k)2655 1450 y Fo(2)2717 1478 y Fu(\000)g Fv(3)p Fl(k)j Fv(+)d(1\))3158 1410 y Fu(\000)3268 1343 y Fv(1)p 3268 1387 49 4 v 3268 1478 a(2)3326 1410 y Fl(:)330 1663 y Fv(Then)34 b Fl(\025)642 1678 y Fo(1)681 1663 y Fl(;)17 b(\025)782 1678 y Fo(2)849 1663 y Fl(>)27 b Fv(0)33 b(and)1409 1880 y Fl(\025)1466 1895 y Fo(1)1505 1880 y Fl(g)t Fv(\()p Fl(\033)1649 1895 y Fp(k)1691 1880 y Fv(\))22 b(+)g Fl(\025)1906 1895 y Fo(2)1946 1880 y Fl(g)t Fv(\()p Fl(\022)2080 1895 y Fp(k)2122 1880 y Fv(\))28 b(=)f Fl(\013)2353 1895 y Fo(1)2415 1880 y Fv(+)22 b Fl(\013)2575 1895 y Fo(2)2642 1880 y Fv(=)28 b(1)p Fl(:)330 2133 y Fv(Comparing)h(this)h(with)g(\(3.8\))g(w)m(e)i (conclude)f(that)g(there)g(are)f(no)h(v)-5 b(alues)30 b Fl(\013)3113 2148 y Fo(1)3183 2133 y Fv(and)h Fl(\013)3433 2148 y Fo(2)3503 2133 y Fv(suc)m(h)h(that)330 2350 y Fl(\013)392 2365 y Fo(1)454 2350 y Fv(+)22 b Fl(\013)614 2365 y Fo(2)681 2350 y Fv(=)27 b(1)33 b(and)1537 2566 y Fl(g)t Fv(\()p Fl(\033)1681 2581 y Fp(k)1723 2566 y Fv(\))p Fl(;)17 b(g)t Fv(\()p Fl(\022)1939 2581 y Fp(k)1981 2566 y Fv(\))28 b Fl(<)2165 2499 y(k)2219 2463 y Fo(2)2280 2499 y Fu(\000)23 b Fv(3)p Fl(k)i Fv(+)d(1)p 2160 2544 496 4 v 2160 2635 a Fl(n)2218 2606 y Fo(2)2280 2635 y Fu(\000)h Fv(3)p Fl(n)f Fv(+)g(1)2666 2566 y Fl(:)p 3829 2820 4 66 v 3833 2757 59 4 v 3833 2820 V 3891 2820 4 66 v 330 3105 a Fm(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(2.2.2.)44 b Fv(Without)36 b(loss)g(of)g(generalit)m(y)-8 b(,)36 b(w)m(e)h(ma)m(y)f(assume)h(that)f(the)h(maxi-)330 3322 y(m)m(um)e(v)-5 b(alue)35 b(of)g Fl(f)997 3337 y Fo(0)1073 3322 y Fv(is)g(attained)g(at)g(the)h(iden)m(tit)m(y)g(p)s (erm)m(utation)e Fl(")i Fv(\(see)h(Remark)e(3.1.2\).)53 b(Ex-)330 3539 y(cluding)37 b(an)i(ob)m(vious)g(case)g(of)f Fl(f)1555 3554 y Fo(0)1633 3539 y Fu(\021)g Fv(0,)i(b)m(y)f(scaling)f Fl(f)11 b Fv(,)39 b(if)f(necessary)-8 b(,)42 b(w)m(e)e(ma)m(y)e(assume) h(that)330 3755 y Fl(f)378 3770 y Fo(0)418 3755 y Fv(\()p Fl(")p Fv(\))32 b(=)g(1.)52 b(Let)36 b Fl(g)j Fv(b)s(e)d(the)f(cen)m (tral)h(pro)5 b(jection)35 b(of)g Fl(f)2332 3770 y Fo(0)2372 3755 y Fv(.)52 b(As)36 b(in)f(the)h(pro)s(of)e(of)i(Theorem)f(2.2.1,) 330 3972 y(w)m(e)f(deduce)g(that)e Fl(g)k Fv(is)c(a)g(linear)f(com)m (bination)g(of)h Fl(\037)2271 3987 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(2)2496 3972 y Fv(and)g Fl(\037)2746 3987 y Fp(n)p Ff(\000)p Fo(2)p Fp(;)p Fo(1)p Fp(;)p Fo(1)3026 3972 y Fv(and)g(that)h Fl(g)t Fv(\()p Fl(")p Fv(\))26 b(=)i(1.)471 4189 y(Let)j(us)h(c)m(ho)s(ose)g(a)f(3)c Fu(\024)h Fl(k)j Fu(\024)d Fl(n)19 b Fu(\000)h Fv(3)31 b(and)g(let)f Fl(X)2186 4204 y Fp(k)2260 4189 y Fv(b)s(e)h(the)g(set)h (of)f(p)s(erm)m(utations)f Fl(\033)35 b Fv(suc)m(h)d(that)330 4405 y Fl(p)p Fv(\()p Fl(\033)t Fv(\))j(=)f Fl(k)40 b Fv(and)d Fl(t)p Fv(\()p Fl(\033)t Fv(\))e(=)g(0)h(and)h(let)f Fl(Y)1741 4420 y Fp(k)1821 4405 y Fv(b)s(e)h(the)g(set)g(of)g(p)s(erm)m (utations)e Fl(\022)40 b Fv(suc)m(h)f(that)d Fl(p)p Fv(\()p Fl(\022)s Fv(\))f(=)g Fl(k)330 4622 y Fv(and)f Fl(t)p Fv(\()p Fl(\022)s Fv(\))d(=)f(1.)48 b(T)-8 b(o)34 b(c)m(ho)s(ose)h(a)f (p)s(erm)m(utation)f Fl(\033)h Fu(2)d Fl(X)2307 4637 y Fp(k)2349 4622 y Fv(,)k(one)f(has)h(to)e(c)m(ho)s(ose)i Fl(k)j Fv(\014xed)d(p)s(oin)m(ts)f(in)330 4758 y Fj(\000)376 4794 y Fp(n)378 4873 y(k)418 4758 y Fj(\001)498 4839 y Fv(w)m(a)m(ys)i(and)e(then)g(a)g(p)s(erm)m(utation)e(without)i (\014xed)h(p)s(oin)m(ts)e(or)h(2-cycles)g(on)g(the)g(remaining)330 5055 y(\()p Fl(n)22 b Fu(\000)h Fl(k)s Fv(\))32 b(p)s(oin)m(ts.)44 b(Then,)33 b(b)m(y)h(Lemma)d(3.4.2)1488 5331 y Fu(j)p Fl(X)1597 5346 y Fp(k)1639 5331 y Fu(j)c(\025)1809 5263 y Fv(1)p 1809 5308 49 4 v 1809 5399 a(5)1868 5190 y Fj(\022)1941 5263 y Fl(n)1944 5399 y(k)2000 5190 y Fj(\023)2073 5331 y Fv(\()p Fl(n)22 b Fu(\000)h Fl(k)s Fv(\)!)28 b(=)2551 5263 y(1)p 2551 5308 V 2551 5399 a(5)2620 5263 y Fl(n)p Fv(!)p 2620 5308 86 4 v 2622 5399 a Fl(k)s Fv(!)2715 5331 y Fl(:)p eop %%Page: 67 75 67 74 bop 2070 -210 a Fw(67)330 91 y Fv(Similarly)-8 b(,)32 b(to)i(c)m(ho)s(ose)i(a)f(p)s(erm)m(utation)e Fl(\022)i Fu(2)d Fl(Y)2079 106 y Fp(k)2122 91 y Fv(,)j(one)g(has)h(to)e (c)m(ho)s(ose)i(a)f(2-cycle)g(in)3500 11 y Fj(\000)3546 47 y Fp(n)3549 126 y Fo(2)3589 11 y Fj(\001)3669 91 y Fv(w)m(a)m(ys,)330 308 y Fl(k)k Fv(\014xed)e(p)s(oin)m(ts)e(in)1072 227 y Fj(\000)1117 264 y Fp(n)p Ff(\000)p Fo(2)1165 342 y Fp(k)1250 227 y Fj(\001)1332 308 y Fv(w)m(a)m(ys)i(and)f(a)f(p)s(erm) m(utation)f(without)i(\014xed)g(p)s(oin)m(ts)g(or)f(2-cycles)h(on)330 525 y(the)d(remaining)d(\()p Fl(n)23 b Fu(\000)f Fl(k)j Fu(\000)e Fv(2\))32 b(p)s(oin)m(ts.)43 b(Then,)34 b(b)m(y)g(Lemma)d (3.4.2)1214 841 y Fu(j)p Fl(Y)1299 856 y Fp(k)1341 841 y Fu(j)d(\025)1512 774 y Fv(1)p 1512 818 49 4 v 1512 909 a(5)1571 701 y Fj(\022)1644 774 y Fl(n)1649 909 y Fv(2)1702 701 y Fj(\023)q(\022)1849 774 y Fl(n)22 b Fu(\000)h Fv(2)1936 909 y Fl(k)2078 701 y Fj(\023)2151 841 y Fv(\()p Fl(n)f Fu(\000)h Fl(k)i Fu(\000)e Fv(2\)!)k(=)2846 774 y Fl(n)p Fv(!)p 2800 818 179 4 v 2800 909 a(10)p Fl(k)s Fv(!)2988 841 y Fl(:)330 1157 y Fv(Let)37 b(us)h(c)m(ho)s(ose)g(a)f(p)s (erm)m(utation)e Fl(\033)40 b Fu(2)c Fl(X)1879 1172 y Fp(k)1958 1157 y Fv(and)h(a)g(p)s(erm)m(utation)f Fl(\022)i Fu(2)e Fl(Y)3043 1172 y Fp(k)3122 1157 y Fv(and)i(let)e Fl(Z)42 b Fv(=)35 b Fl(X)3763 1172 y Fp(k)3843 1157 y Fv(if)330 1374 y Fl(g)t Fv(\()p Fl(\033)t Fv(\))27 b Fu(\025)h Fl(g)t Fv(\()p Fl(\022)s Fv(\))k(and)h Fl(Z)h Fv(=)28 b Fl(Y)1307 1389 y Fp(k)1381 1374 y Fv(otherwise.)44 b(Then)1885 1690 y Fu(j)p Fl(Z)7 b Fu(j)27 b(\025)2203 1623 y Fl(n)p Fv(!)p 2157 1667 V 2157 1759 a(10)p Fl(k)s Fv(!)330 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b(in)f(the)i(pro)s(of)e(of)g(Theorem)h(2.3.2,)330 1608 y(w)m(e)g(ha)m(v)m(e)1859 1825 y Fu(j)p Fl(X)1968 1840 y Fp(k)2010 1825 y Fu(j)27 b(\025)2180 1757 y Fv(1)p 2180 1802 49 4 v 2180 1893 a(5)2249 1757 y Fl(n)p Fv(!)p 2249 1802 86 4 v 2251 1893 a Fl(k)s Fv(!)2344 1825 y Fl(:)330 2091 y Fv(T)-8 b(o)41 b(c)m(ho)s(ose)h(a)f(p)s(erm)m(utation)e Fl(\021)46 b Fu(2)c Fl(Y)1715 2106 y Fp(k)1757 2091 y Fv(,)h(one)f(has)f(to)g(c)m(ho)s(ose)h Fl(k)i Fv(transp)s(ositions)39 b(\(2-cycles\))j(in)577 2240 y Fl(n)p Fv(!)p 340 2284 559 4 v 340 2376 a(\()p Fl(n)22 b Fu(\000)h Fv(2)p Fl(k)s Fv(\)!)p Fl(k)s Fv(!2)856 2347 y Fp(k)950 2307 y Fv(w)m(a)m(ys)43 b(and)e(a)g(p)s(erm)m(utation)f(without)g(\014xed)j(p)s(oin)m(ts)e(or)g (2-cycles)g(on)g(the)h(re-)330 2524 y(maining)30 b Fl(n)22 b Fu(\000)h Fv(2)p Fl(k)35 b Fv(p)s(oin)m(ts.)43 b(Hence,)34 b(b)m(y)g(Lemma)d(3.4.2)1827 2829 y Fu(j)p Fl(Y)1912 2844 y Fp(k)1954 2829 y Fu(j)c(\025)2124 2761 y Fv(1)p 2124 2806 49 4 v 2124 2897 a(5)2237 2761 y Fl(n)p 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Fl(Z)r(:)330 4569 y Fv(The)34 b(pro)s(of)d(no)m(w)i(pro)s (ceed)h(as)f(in)e(the)i(pro)s(of)f(of)g(Theorem)h(2.2.2,)f(Section)h (3.6.)p 3829 4569 4 66 v 3833 4506 59 4 v 3833 4569 V 3891 4569 4 66 v 330 4858 a Fm(R)-5 b(emark)34 b(3.7.3)g(\(Sc)-5 b(ar)g(city)35 b(of)g(r)-5 b(elatively)34 b(go)-5 b(o)g(d)34 b(values\).)49 b Fv(Let)38 b(us)g(consider)g(the)h(function)e Fl(f)48 b Fv(of)330 5075 y(Section)32 b(2.3.2.)43 b(W)-8 b(e)33 b(observ)m(e)i(that)1745 5380 y Fl(f)k Fv(=)27 b Fl(\013)1997 5395 y Fo(1)2037 5380 y Fl(r)2081 5395 y Fo(3)2142 5380 y Fv(+)22 b Fl(\013)2302 5395 y Fo(2)2342 5380 y Fl(r)2386 5395 y Fo(7)p Fp(e)2458 5380 y Fl(;)p eop %%Page: 72 80 72 79 bop 2070 -210 a Fw(72)330 91 y Fv(for)681 308 y Fl(\013)743 323 y Fo(1)810 308 y Fv(=)924 241 y Fl(n)982 204 y Fo(2)1044 241 y Fu(\000)22 b Fl(nm)h Fu(\000)g Fv(2)p Fl(n)f Fv(+)g(3)p Fl(m)g Fu(\000)h Fv(3)p 924 285 1017 4 v 949 376 a Fl(n)1007 348 y Fo(2)1069 376 y Fu(\000)g Fl(nm)f Fv(+)g(2)p Fl(n)g Fv(+)g Fl(m)h Fu(\000)g Fv(5)2048 308 y(and)98 b Fl(\013)2365 323 y Fo(2)2432 308 y Fv(=)2762 241 y(4)p Fl(n)22 b Fu(\000)h Fv(2)p Fl(m)f Fu(\000)g Fv(2)p 2545 285 967 4 v 2545 376 a Fl(n)2603 348 y Fo(2)2665 376 y Fu(\000)h Fl(nm)g Fv(+)f(2)p Fl(n)g Fv(+)g Fl(m)g Fu(\000)h Fv(5)3522 308 y Fl(:)330 567 y Fv(Th)m(us)35 b Fl(f)44 b Fv(is)33 b(a)g(con)m(v)m(ex)j(com)m (bination)31 b(of)i Fl(r)1884 582 y Fo(3)1957 567 y Fv(and)h Fl(r)2192 582 y Fo(7)p Fp(e)2264 567 y Fv(,)g(hence)h(1)29 b(=)g Fl(f)11 b Fv(\()p Fl(")p Fv(\))28 b Fu(\025)i Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))33 b(for)g(all)e Fl(\033)i Fu(2)d Fl(S)3853 582 y Fp(n)330 783 y Fv(and)p 513 702 59 4 v 26 w Fl(f)38 b Fv(=)28 b(0.)41 b(Remark)25 b(3.2.5)h(implies)d (that)j Fl(f)36 b Fv(is)25 b(a)h(generalized)f(function)g(\(1.2\))h(of) f(the)h(required)330 1000 y(t)m(yp)s(e.)330 1262 y Ft(3.8)112 b(Pro)s(of)37 b(of)h(General)f(Case)471 1578 y Fv(The)51 b(pro)s(of)d(of)h(Theorem)h(2.4.1)f(follo)m(ws)f(from)h(our)g (description)g(of)g(the)h(cen)m(tral)f(cone)330 1794 y(\(Lemma)44 b(3.3.1\),)k(and)d(the)h(observ)-5 b(ations)45 b(used)i(in)d(the)i(pro)s(of)f(of)g(Theorem)g(2.2.1)g(in)g(Sec-)330 2011 y(tion)32 b(3.6.)471 2228 y(T)-8 b(o)38 b(pro)m(v)m(e)g(Theorem)g (2.4.2,)g(w)m(e)h(need)f(another)g(lemma)d(sho)m(wing)j(that)f(the)h (cen)m(tral)g(pro-)330 2444 y(jection)32 b(has)h(go)s(o)s(d)e(v)-5 b(alues)33 b(on)f(at)h(least)f(one)h(of)f(sev)m(eral)h(large)e(classes) j(of)e(p)s(erm)m(utations.)330 2716 y Ft(Lemma)37 b(3.8.1.)147 b Fd(Let)43 b Fl(g)j Fd(b)s(e)d(a)g(linear)e(com)m(bination)g(of)h Fl(g)2663 2731 y Fo(1)2748 2716 y Fv(=)j Fl(p)29 b Fu(\000)g Fv(1)p Fd(,)46 b Fl(g)3222 2731 y Fo(2)3306 2716 y Fv(=)f Fl(p)3476 2680 y Fo(2)3545 2716 y Fu(\000)29 b Fv(2)43 b Fd(and)330 2933 y Fl(g)377 2948 y Fo(3)467 2933 y Fv(=)50 b(1)32 b Fu(\000)f Fv(2)p Fl(t)47 b Fd(suc)m(h)g(that)f(suc)m(h)i(that) e Fl(g)t Fv(\()p Fl(")p Fv(\))j(=)i(1)p Fd(.)84 b(F)-8 b(or)45 b(a)h Fv(2)k Fu(\024)h Fl(k)j Fu(\024)d Fl(n)32 b Fu(\000)g Fv(2)p Fd(,)49 b(let)d Fl(\033)3616 2948 y Fp(k)3705 2933 y Fd(b)s(e)g(a)330 3149 y(p)s(erm)m(utation)37 b(suc)m(h)i(that)f Fl(p)p Fv(\()p Fl(\033)1478 3164 y Fp(k)1521 3149 y Fv(\))f(=)g Fl(k)k Fd(and)d Fl(t)p Fv(\()p Fl(\033)2124 3164 y Fp(k)2168 3149 y Fv(\))f(=)f(0)p Fd(,)k(let)d Fl(\021)42 b Fd(b)s(e)c(a)g(p)s(erm)m(utation)f(suc)m(h)i (that)330 3366 y Fl(p)p Fv(\()p Fl(\021)t Fv(\))27 b(=)h(0)k Fd(and)h Fl(t)p Fv(\()p Fl(\021)t Fv(\))27 b(=)h(1)k Fd(and)h(let)f Fl(\022)j Fd(b)s(e)e(p)s(erm)m(utation)e(suc)m(h)j(that) f Fl(p)p Fv(\()p Fl(\022)s Fv(\))27 b(=)h Fl(t)p Fv(\()p Fl(\022)s Fv(\))g(=)f(0)p Fd(.)44 b(Then)1206 3654 y Fv(max)1388 3574 y Fj(\010)1446 3654 y Fl(g)t Fv(\()p Fl(\033)1590 3669 y Fp(k)1632 3654 y Fv(\))p Fl(;)17 b(g)t Fv(\()p Fl(\021)t Fv(\))p Fl(;)g(g)t Fv(\()p Fl(\022)s Fv(\))2112 3574 y Fj(\011)2195 3654 y Fu(\025)2536 3587 y Fl(k)26 b Fu(\000)c Fv(2)p 2310 3631 677 4 v 2310 3723 a Fl(n)2368 3694 y Fo(2)2430 3723 y Fu(\000)h Fl(k)s(n)f Fv(+)g Fl(k)k Fu(\000)c Fv(2)2997 3654 y Fl(:)330 3943 y Fm(Pr)-5 b(o)g(of.)45 b Fv(W)-8 b(e)33 b(can)g(write)1318 4231 y Fl(g)e Fv(=)d Fl(\013)1562 4246 y Fo(1)1616 4164 y Fl(p)22 b Fu(\000)h Fv(1)p 1611 4208 229 4 v 1611 4299 a Fl(n)g Fu(\000)f Fv(1)1872 4231 y(+)g Fl(\013)2032 4246 y Fo(2)2086 4164 y Fl(p)2135 4127 y Fo(2)2197 4164 y Fu(\000)g Fv(2)p 2082 4208 269 4 v 2082 4299 a Fl(n)2140 4271 y Fo(2)2201 4299 y Fu(\000)h Fv(2)2382 4231 y(+)f Fl(\013)2542 4246 y Fo(3)2581 4231 y Fv(\(1)g Fu(\000)h Fv(2)p Fl(t)p Fv(\))330 4519 y(for)32 b(some)g Fl(\013)785 4534 y Fo(1)825 4519 y Fl(;)17 b(\013)931 4534 y Fo(2)1003 4519 y Fv(and)32 b Fl(\013)1254 4534 y Fo(3)1326 4519 y Fv(suc)m(h)i(that)f Fl(\013)1820 4534 y Fo(1)1881 4519 y Fv(+)22 b Fl(\013)2041 4534 y Fo(2)2103 4519 y Fv(+)g Fl(\013)2263 4534 y Fo(3)2330 4519 y Fv(=)28 b(1.)43 b(Then)1438 4770 y Fl(g)t Fv(\()p Fl(\033)1582 4785 y Fp(k)1624 4770 y Fv(\))28 b(=)f Fl(\013)1855 4785 y Fo(1)1907 4703 y Fl(k)e Fu(\000)e Fv(1)p 1905 4747 229 4 v 1905 4839 a Fl(n)f Fu(\000)h Fv(1)2166 4770 y(+)f Fl(\013)2326 4785 y Fo(2)2377 4703 y Fl(k)2431 4667 y Fo(2)2493 4703 y Fu(\000)h Fv(2)p 2375 4747 269 4 v 2375 4839 a Fl(n)2433 4810 y Fo(2)2495 4839 y Fu(\000)g Fv(2)2676 4770 y(+)f Fl(\013)2836 4785 y Fo(3)1438 5060 y Fl(g)t Fv(\()p Fl(\021)t Fv(\))k(=)i Fu(\000)1898 4992 y Fl(\013)1960 5007 y Fo(1)p 1834 5037 229 4 v 1834 5128 a Fl(n)23 b Fu(\000)f Fv(1)2095 5060 y Fu(\000)h Fl(\013)2257 5075 y Fo(2)2416 4992 y Fv(2)p 2306 5037 269 4 v 2306 5128 a Fl(n)2364 5099 y Fo(2)2426 5128 y Fu(\000)g Fv(2)2607 5060 y Fu(\000)f Fl(\013)2768 5075 y Fo(3)1438 5349 y Fl(g)t Fv(\()p Fl(\022)s Fv(\))27 b(=)h Fu(\000)1895 5281 y Fl(\013)1957 5296 y Fo(1)p 1831 5326 229 4 v 1831 5417 a Fl(n)22 b Fu(\000)h Fv(1)2092 5349 y Fu(\000)f Fl(\013)2253 5364 y Fo(2)2413 5281 y Fv(2)p 2303 5326 269 4 v 2303 5417 a Fl(n)2361 5388 y Fo(2)2423 5417 y Fu(\000)g Fv(2)2603 5349 y(+)g Fl(\013)2763 5364 y Fo(3)2803 5349 y Fl(:)p eop %%Page: 73 81 73 80 bop 2070 -210 a Fw(73)330 91 y Fv(W)-8 b(e)34 b(observ)m(e)h (that)f Fl(g)t Fv(\()p Fl(\033)1204 106 y Fp(k)1246 91 y Fv(\),)g Fl(g)t Fv(\()p Fl(\021)t Fv(\))e(and)i Fl(g)t Fv(\()p Fl(\022)s Fv(\))f(are)h(linear)e(functions)h(of)g Fl(\013)2986 106 y Fo(1)3026 91 y Fl(;)17 b(\013)3132 106 y Fo(2)3205 91 y Fv(and)33 b Fl(\013)3457 106 y Fo(3)3530 91 y Fv(and)h(that)330 308 y(for)798 525 y Fl(\013)860 540 y Fo(1)927 525 y Fv(=)1200 457 y Fl(k)s Fv(\(1)22 b Fu(\000)g Fl(n)p Fv(\))p 1041 502 677 4 v 1041 593 a Fl(n)1099 564 y Fo(2)1161 593 y Fu(\000)g Fl(nk)k Fv(+)c Fl(k)j Fu(\000)e Fv(2)1727 525 y Fl(;)115 b(\013)1931 540 y Fo(2)1998 525 y Fv(=)2316 457 y Fl(n)2374 421 y Fo(2)2435 457 y Fu(\000)23 b Fv(2)p 2111 502 V 2111 593 a Fl(n)2169 564 y Fo(2)2231 593 y Fu(\000)g Fl(nk)i Fv(+)d Fl(k)k Fu(\000)c Fv(2)2895 525 y(and)98 b Fl(\013)3212 540 y Fo(3)3279 525 y Fv(=)28 b(0)330 795 y(w)m(e)34 b(ha)m(v)m(e)605 1112 y Fl(g)t Fv(\()p Fl(\033)749 1127 y Fp(k)791 1112 y Fv(\))28 b(=)f Fl(g)t Fv(\()p Fl(\021)t Fv(\))g(=)h Fl(g)t Fv(\()p Fl(\022)s Fv(\))f(=)1811 1044 y Fl(k)e Fu(\000)e Fv(2)p 1585 1089 V 1585 1180 a Fl(n)1643 1151 y Fo(2)1705 1180 y Fu(\000)g Fl(nk)i Fv(+)d Fl(k)j Fu(\000)e Fv(2)2369 1112 y(and)98 b Fl(\013)2686 1127 y Fo(1)2747 1112 y Fv(+)22 b Fl(\013)2907 1127 y Fo(2)2969 1112 y Fv(+)g Fl(\013)3129 1127 y Fo(3)3196 1112 y Fv(=)28 b(1)p Fl(:)275 b Fv(\(3.10\))330 1428 y(Let)763 1645 y Fl(\025)820 1660 y Fo(1)887 1645 y Fv(=)1000 1577 y Fl(n)1058 1541 y Fo(2)1120 1577 y Fu(\000)23 b Fv(2)p Fl(n)p 1000 1622 327 4 v 1004 1713 a(k)1058 1684 y Fo(2)1120 1713 y Fu(\000)g Fv(2)p Fl(k)1337 1645 y(;)114 b(\025)1535 1660 y Fo(2)1602 1645 y Fv(=)1715 1577 y Fl(n)1773 1541 y Fo(2)1835 1577 y Fu(\000)23 b Fl(nk)p 1715 1622 332 4 v 1744 1713 a Fv(2)p Fl(k)i Fu(\000)e Fv(4)2155 1645 y(and)97 b Fl(\025)2466 1660 y Fo(3)2533 1645 y Fv(=)2647 1577 y Fl(n)2705 1541 y Fo(2)2767 1577 y Fu(\000)22 b Fl(k)s(n)h Fu(\000)f Fv(2)p Fl(n)g Fv(+)g(2)p Fl(k)p 2647 1622 784 4 v 2987 1713 a Fv(2)p Fl(k)3440 1645 y(:)330 1915 y Fv(Then)34 b Fl(\025)642 1930 y Fo(1)681 1915 y Fl(;)17 b(\025)782 1930 y Fo(2)821 1915 y Fl(;)g(\025)922 1930 y Fo(3)989 1915 y Fl(>)27 b Fv(0)33 b(and)1121 2232 y Fl(\025)1178 2247 y Fo(1)1217 2232 y Fl(g)t Fv(\()p Fl(\033)1361 2247 y Fp(k)1403 2232 y Fv(\))22 b(+)g Fl(\025)1618 2247 y Fo(2)1658 2232 y Fl(g)t Fv(\()p Fl(\021)t Fv(\))f(+)h Fl(\025)2013 2247 y Fo(3)2052 2232 y Fl(g)t Fv(\()p Fl(\022)s Fv(\))27 b(=)h Fl(\013)2420 2247 y Fo(1)2481 2232 y Fv(+)22 b Fl(\013)2641 2247 y Fo(2)2703 2232 y Fv(+)g Fl(\013)2863 2247 y Fo(3)2930 2232 y Fv(=)28 b(1)p Fl(:)330 2548 y Fv(Comparing)g(this)h(with)h(\(3.10\),)f(w)m(e)i(conclude)f(that)g (there)g(are)g(no)g(v)-5 b(alues)29 b Fl(\013)3180 2563 y Fo(1)3220 2548 y Fl(;)17 b(\013)3326 2563 y Fo(2)3394 2548 y Fv(and)30 b Fl(\013)3643 2563 y Fo(3)3712 2548 y Fv(suc)m(h)330 2765 y(that)i Fl(\013)603 2780 y Fo(1)665 2765 y Fv(+)22 b Fl(\013)825 2780 y Fo(2)887 2765 y Fv(+)g Fl(\013)1047 2780 y Fo(3)1114 2765 y Fv(=)27 b(1)33 b(and)1356 3081 y Fl(g)t Fv(\()p Fl(\033)1500 3096 y Fp(k)1542 3081 y Fv(\))p Fl(;)17 b(g)t Fv(\()p Fl(\021)t Fv(\))p Fl(;)g(g)t Fv(\()p Fl(\022)s Fv(\))25 b Fl(<)2387 3014 y(k)g Fu(\000)d Fv(2)p 2161 3058 677 4 v 2161 3149 a Fl(n)2219 3121 y Fo(2)2281 3149 y Fu(\000)g Fl(nk)k Fv(+)c Fl(k)j Fu(\000)d Fv(2)2847 3081 y Fl(:)p 3829 3397 4 66 v 3833 3335 59 4 v 3833 3397 V 3891 3397 4 66 v 330 3689 a Fm(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(2.4.2.)44 b Fv(The)32 b(pro)s(of)f(follo)m(ws)e(those)j(of)f(Theorem)g(2.2.2)f (\(Section)h(3.6\))g(and)330 3905 y(Theorem)36 b(2.3.2)f(\(Section)g (3.7\))g(with)g(some)g(mo)s(di\014cations.)50 b(Let)35 b Fl(X)2937 3920 y Fp(k)3015 3905 y Fv(b)s(e)h(the)g(set)g(of)f(all)e (p)s(er-)330 4122 y(m)m(utations)f Fl(\033)k Fv(suc)m(h)e(that)f Fl(p)p Fv(\()p Fl(\033)t Fv(\))27 b(=)h Fl(k)36 b Fv(and)c Fl(t)p Fv(\()p Fl(\033)t Fv(\))c(=)g(0.)44 b(As)33 b(in)f(the)h(pro)s (of)e(of)i(Theorem)g(2.2.2,)f(w)m(e)330 4339 y(ha)m(v)m(e)1859 4555 y Fu(j)p Fl(X)1968 4570 y Fp(k)2010 4555 y Fu(j)27 b(\025)2180 4488 y Fv(1)p 2180 4533 49 4 v 2180 4624 a(5)2249 4488 y Fl(n)p Fv(!)p 2249 4533 86 4 v 2251 4624 a Fl(k)s Fv(!)2344 4555 y Fl(:)330 4826 y Fv(Let)34 b Fl(Y)56 b Fv(b)s(e)34 b(the)h(set)g(of)e(all)f(p)s(erm)m(utations)i Fl(\033)k Fv(suc)m(h)d(that)f Fl(p)p Fv(\()p Fl(\033)t Fv(\))d(=)f(0)k(and)g Fl(t)p Fv(\()p Fl(\033)t Fv(\))d(=)f(1.)48 b(T)-8 b(o)34 b(c)m(ho)s(ose)330 5043 y(a)d(p)s(erm)m(utation)f Fl(\033)i Fu(2)c Fl(Y)21 b Fv(,)31 b(one)h(has)g(to)f(c)m(ho)s(ose)h(a) f(2-cycle)g(in)2574 4962 y Fj(\000)2620 4999 y Fp(n)2624 5077 y Fo(2)2663 4962 y Fj(\001)2740 5043 y Fv(w)m(a)m(ys)h(and)g(then) g(an)f(arbitrary)330 5259 y(p)s(erm)m(utation)43 b(of)g(the)i (remaining)d(\()p Fl(n)30 b Fu(\000)g Fv(2\))44 b(sym)m(b)s(ols)g (without)f(\014xed)j(p)s(oin)m(ts)d(and)h(2-cycles.)p eop %%Page: 74 82 74 81 bop 2070 -210 a Fw(74)330 91 y Fv(Using)32 b(Lemma)f(3.4.2,)i(w)m (e)g(estimate)1401 406 y Fu(j)p Fl(Y)21 b Fu(j)27 b(\025)1677 339 y Fv(1)p 1677 383 49 4 v 1677 475 a(5)1893 339 y Fl(n)p Fv(!)p 1746 383 381 4 v 1746 475 a(2\()p Fl(n)22 b Fu(\000)h Fv(2\)!)2136 406 y(\()p Fl(n)g Fu(\000)f Fv(2\)!)28 b(=)2634 339 y(1)p 2609 383 98 4 v 2609 475 a(10)2717 406 y Fl(n)p Fv(!)p Fl(:)330 721 y Fv(Let)44 b(us)g(c)m(ho)s(ose)g(a)f(p)s(erm)m(utation)g Fl(\033)1689 736 y Fp(k)1778 721 y Fu(2)k Fl(X)1972 736 y Fp(k)2014 721 y Fv(,)g(a)c(p)s(erm)m(utation)f Fl(\021)50 b Fu(2)d Fl(Y)64 b Fv(and)44 b(a)f(p)s(erm)m(utation)330 938 y Fl(\022)31 b Fu(2)d Fl(X)581 953 y Fo(0)620 938 y Fv(.)43 b(Let)29 b(us)g(c)m(ho)s(ose)h Fl(Z)36 b Fv(to)29 b(b)s(e)g(one)g(of)g Fl(X)2000 953 y Fp(k)2042 938 y Fv(,)h Fl(X)2180 953 y Fo(0)2248 938 y Fv(and)f Fl(Y)22 b Fv(,)29 b(dep)s(ending)h(where)g (the)f(maxim)m(um)330 1155 y(v)-5 b(alue)32 b(of)g Fl(g)t Fv(\()p Fl(\033)837 1170 y Fp(k)879 1155 y Fv(\),)h Fl(g)t Fv(\()p Fl(\021)t Fv(\))e(or)i Fl(g)t Fv(\()p Fl(\022)s Fv(\))e(is)i(attained.)42 b(Hence)1909 1470 y Fu(j)p Fl(Z)7 b 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b(This)33 b(is)f(a)h(maxim)m(um)d(of)i Fl(f)43 b Fv(if)32 b(and)g(only)g(if:)1869 4859 y Fl(b;)17 b(c;)g(d;)g(e)27 b Fu(\024)h Fv(1)1290 b(\(3.12\))330 5163 y(In)38 b(fact,)g(since)g Fl(A)g Fv(is)f(constan)m(t,)j(the)e(pro) 5 b(jection)37 b Fl(g)k Fv(of)c(the)h Fk(QAP)g Fv(\()p Fl(A;)17 b(B)5 b Fv(\))37 b(is)g(a)g(linear)f(function)330 5380 y(of)h(the)i(parameters)f Fl(b;)17 b(c;)g(d;)g(e)p Fv(.)59 b(Th)m(us)39 b(Equations)g(3.11)e(and)h(3.12)f(de\014ne)i(a)f (p)s(olytop)s(e)f(in)g(the)p eop %%Page: 77 85 77 84 bop 2070 -210 a Fw(77)330 91 y Fv(parameter)38 b(space.)63 b(The)40 b(cen)m(tral)e(pro)5 b(jection)38 b(of)h(the)g(set)g(of)f Fk(QAP)p Fv('s)i(\()p Fl(A;)17 b(B)5 b Fv(\))39 b(de\014ned)h(b)m(y)f(this)330 308 y(p)s(olytop)s(e)28 b(is)g(the)h(set)g(of)f Fl(g)k Fv(attainable)26 b(from)i Fk(QAP)p Fv('s)h(of)f(the)h(ab)s(o)m(v)m(e)h(form.)41 b(Since)28 b(the)h(maxim)m(um)330 525 y(is)i(1)h(and)f(the)i(a)m(v)m (erage)f(is)f(0,)h(this)g(pro)5 b(jection)31 b(is)g(a)h(con)m(v)m(ex)i 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y(\(1.4\))25 b(is)g(spanned)i(b)m(y)f(the)g(c)m(haracters)g Fl(\037)1826 4634 y Fp(n)1899 4619 y Fv(and)f Fl(\037)2142 4634 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2360 4619 y Fv(from)f(Section)h(3.2.)41 b(So)25 b(if)g(w)m(e)h(prepro)s(cess)330 4836 y(the)34 b(function)e Fl(f)44 b Fv(\(cf.)h(Remark)33 b(3.1.2\))g(so)g(that)g(it)f(has)i(a)m(v)m(erage)g(v)-5 b(alue)32 b(0)h(and)h(maxim)m(um)c(1)j(at)330 5053 y(the)g(iden)m(tit)m (y)f Fl(")p Fv(,)h(the)g(cen)m(tral)f(pro)5 b(jection)32 b Fl(g)k Fv(of)c Fl(f)44 b Fv(around)32 b(this)g(maxim)m(um)f(will)f(b) s(e)j(:)1500 5349 y Fl(g)t Fv(\()p Fl(\033)t Fv(\))27 b(=)1826 5281 y Fl(\037)1887 5296 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)2080 5281 y Fv(\()p Fl(\033)t Fv(\))p 1826 5326 389 4 v 1906 5417 a Fl(n)22 b Fu(\000)h Fv(1)2252 5349 y(=)2366 5281 y Fl(p)p Fv(\()p Fl(\033)t Fv(\))f Fu(\000)g Fv(1)p 2366 5326 355 4 v 2428 5417 a Fl(n)h Fu(\000)f Fv(1)2070 5680 y Fw(79)p eop %%Page: 80 88 80 87 bop 2070 -210 a Fw(80)330 91 y Fv(where)44 b Fl(p)p Fv(\()p Fl(\033)t Fv(\))e(is)g(the)h(n)m(um)m(b)s(er)g(of)f(\014xed)h (p)s(oin)m(ts)f(in)g Fl(\033)k Fv(\(see)e(De\014nition)d(1.1.9\).)72 b(Then)44 b Fl(g)i Fv(has)330 308 y(the)40 b(\\bullsey)m(e"-t)m(yp)s(e) g(distribution)e(that)h(w)m(e)i(suggest)g(is)e(c)m(haracteristic)g(of)g (relativ)m(ely)g(easy)330 525 y(optimization)29 b(problems.)471 741 y(W)-8 b(e)31 b(noted)h(in)e(De\014nition)f(1.1.4)h(that)h(if)f (the)h(cen)m(tral)g(pro)5 b(jection)31 b(of)f(a)h Fk(QAP)g Fv(is)g(spanned)h(b)m(y)330 958 y(the)39 b(c)m(haracters)h Fl(\037)1035 973 y Fp(n)1120 958 y Fv(and)f Fl(\037)1377 973 y Fp(n)p Ff(\000)p Fo(1)p Fp(;)p Fo(1)1569 958 y Fv(,)h(then)f(that)f Fk(QAP)h Fv(reduces)h(to)e(a)g Fk(LAP)p Fv(,)i(whic)m(h)f(is)e(solv)-5 b(able)38 b(in)330 1175 y(p)s(olynomial)29 b(time.)330 1427 y Fs(4.1.2)105 b (Higher-dimensional)35 b(Assignmen)m(t)g(Problems)471 1708 y Fv(W)-8 b(e)32 b(can)g(also)f(consider)h(the)g(higher)f (dimensional)e(assignmen)m(t)i(problems)g(and)h(their)f(sp)s(e-)330 1925 y(cial)j(cases.)55 b(The)37 b Fl(k)s Fv(-dimensional)c(assignmen)m (t)i(problem)g([La)m(w63])h(is)f(the)i(problem)d(of)i(maxi-)330 2141 y(mizing)30 b(for)i(some)h(2)p Fl(k)s Fv(-dimensional)c(tensor)k Fl(C)40 b Fv(the)33 b(function:)1456 2458 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))28 b(=)1959 2333 y Fp(n)1909 2363 y Fj(X)1781 2573 y Fp(i)1805 2582 y Fg(1)1840 2573 y Fp(;i)1884 2582 y Fg(2)1918 2573 y Fp(;:::)10 b(;i)2052 2585 y Fe(k)2090 2573 y Fo(=1)2197 2458 y Fl(c)2239 2412 y Fp(i)2263 2421 y Fg(1)2297 2412 y Fp(i)2321 2421 y Fg(2)2356 2412 y Fp(:::)o(i)2439 2424 y Fe(k)2239 2489 y Fp(\033)r Fo(\()p Fp(i)2332 2498 y Fg(1)2367 2489 y Fo(\))p Fp(\033)r Fo(\()p Fp(i)2487 2498 y Fg(2)2524 2489 y Fo(\))p Fp(:::)o(\033)r Fo(\()p Fp(i)2703 2501 y Fe(k)2742 2489 y Fo(\))3699 2458 y Fv(\(4.1\))330 2786 y(Then)32 b(the)f(cen)m(tral)g(pro)5 b(jection)31 b Fl(g)j Fv(lies)29 b(in)i(2)1941 2750 y Fp(k)1983 2786 y Fv(-dimensional)d (subspace)33 b(spanned)f(b)m(y)g(c)m(haracters)330 3003 y(that)j(w)m(e)i(could)e(in)f(principle)g(\014nd)i(explicitly)-8 b(.)51 b(The)36 b(di\016cult)m(y)f(lies)g(in)f(w)m(orking)i(in)f(this)g (high)330 3220 y(dimensional)30 b(space.)471 3436 y(W)-8 b(e)38 b(cite)f(t)m(w)m(o)h(examples)f(of)g(higher)g(dimensional)e (assignmen)m(t)i(problems)g(that)g(are)g(con-)330 3653 y(sidered)c(in)m(teresting)f(optimization)d(mo)s(dels.)330 3905 y Fs(4.1.3)105 b(The)35 b(BiQuadratic)g(Assignmen)m(t)f(Problem) 471 4186 y Fv(The)51 b Fm(BiQuadr)-5 b(atic)50 b(Assignment)g(Pr)-5 b(oblem)49 b Fv(\()p Fk(BiQAP)p Fv(\))i(is)f(a)f(4-dimensional)e (assignmen)m(t)330 4403 y(problem,)42 b(where)h(the)e (\(8-dimensional\))d(tensor)k Fl(C)48 b Fv(decomp)s(oses)42 b(as)g(a)f(pro)s(duct)g(of)g(t)m(w)m(o)h(\(4-)330 4620 y(dimensional\))34 b(tensors.)56 b(That)36 b(is,)h(for)f(some)g Fl(A;)17 b(B)39 b Fu(2)c Fi(R)2465 4584 y Fo(4)2510 4620 y Fv(,)j(the)e(tensor)h Fl(C)44 b Fv(in)35 b(Equation)h(\(4.1\))330 4836 y(is:)1606 5053 y Fl(c)1648 5010 y Fp(i)1672 5019 y Fg(1)1707 5010 y Fp(i)1731 5019 y Fg(2)1765 5010 y Fp(i)1789 5019 y Fg(3)1824 5010 y Fp(i)1848 5019 y Fg(4)1648 5079 y Fp(j)1677 5088 y Fg(1)1711 5079 y Fp(j)1740 5088 y Fg(2)1774 5079 y Fp(j)1803 5088 y Fg(3)1837 5079 y Fp(j)1866 5088 y Fg(4)1932 5053 y Fv(=)28 b Fl(a)2087 5068 y Fp(i)2111 5077 y Fg(1)2146 5068 y Fp(i)2170 5077 y Fg(2)2204 5068 y Fp(i)2228 5077 y Fg(3)2263 5068 y Fp(i)2287 5077 y Fg(4)2326 5053 y Fl(b)2367 5068 y Fp(j)2396 5077 y Fg(1)2430 5068 y Fp(j)2459 5077 y Fg(2)2493 5068 y Fp(j)2522 5077 y Fg(3)2557 5068 y Fp(j)2586 5077 y Fg(4)p eop %%Page: 81 89 81 88 bop 2070 -210 a Fw(81)330 91 y Fv(This)43 b(problem)g(arises)g (in)f(V)-8 b(ery)44 b(Large)g(Scale)f(In)m(tegrated)h(circuit)e(design) h(\(VLSI\),)h(see)g(for)330 308 y(example)32 b([BC)-56 b(\030)14 b(K94)o(].)330 560 y Fs(4.1.4)105 b(The)35 b(W)-9 b(eigh)m(ted)35 b(Hyp)s(ergraph)g(Matc)m(hing)h(Problem)471 841 y Fv(In)g(the)f Fm(Weighte)-5 b(d)37 b(Hyp)-5 b(er)g(gr)g(aph)38 b(Matching)f(Pr)-5 b(oblem)34 b Fv(w)m(e)j(are)e(giv)m(en)g(a)g Fl(k)s Fv(-uniform)e(h)m(yp)s(er-)330 1057 y(graph)j(on)h(a)f(set)h(of) f Fl(n)g Fv(v)m(ertices.)56 b(The)38 b Fm(e)-5 b(dges)35 b Fv(of)h(suc)m(h)i(a)e(h)m(yp)s(ergraph)h(are)g(subsets)h(of)e(size)h Fl(k)s Fv(;)330 1274 y(eac)m(h)25 b(edge)g Fl(e)800 1289 y Fp(i)852 1274 y Fv(has)g(an)f(asso)s(ciated)g(w)m(eigh)m(t)g Fl(w)1971 1289 y Fp(i)1999 1274 y Fv(.)41 b(The)25 b(problem)e(is)h(to) f(\014nd)i(the)g(maxim)m(um)d(w)m(eigh)m(t)330 1491 y Fm(matching)p Fv(,)39 b(or)g(set)g(of)g(disjoin)m(t)e(edges.)64 b(Some)38 b(applications)f(of)h(this)g(problem)g(are)h(found)g(in)330 1708 y([V)-8 b(em98)o(].)471 1924 y(W)g(e)53 b(can)f(reduce)i(W)-8 b(eigh)m(ted)52 b(Hyp)s(ergraph)h(Matc)m(hing)f(to)g Fl(k)s Fv(-dimensional)d(assignmen)m(t)330 2141 y(m)m(uc)m(h)44 b(as)f(w)m(e)i(reduced)g Fk(TSP)f Fv(to)f Fk(QAP)p Fv(.)h(W)-8 b(e)44 b(enco)s(de)g(the)g(incidence)f(matrix)f(of)h(a)g(maxim)m(um)330 2358 y(matc)m(hing)31 b(in)h Fl(A)c Fv(=)g(\()p Fl(a)1163 2373 y Fp(i)1187 2382 y Fg(1)1221 2373 y Fp(i)1245 2382 y Fg(2)1280 2373 y Fp(:::i)1364 2385 y Fe(k)1405 2358 y Fv(\):)446 2782 y Fl(a)497 2797 y Fp(i)521 2806 y Fg(1)556 2797 y Fp(i)580 2806 y Fg(2)614 2797 y Fp(:::)o(i)697 2809 y Fe(k)767 2782 y Fv(=)871 2488 y Fj(8)871 2578 y(>)871 2608 y(>)871 2638 y(>)871 2668 y(<)871 2847 y(>)871 2877 y(>)871 2907 y(>)871 2937 y(:)959 2679 y Fv(1)97 b(if)64 b Fu(f)p Fl(i)1310 2694 y Fo(1)1350 2679 y Fl(;)17 b(:)g(:)g(:)32 b(;)17 b(i)1618 2694 y Fp(k)1661 2679 y Fu(g)27 b Fv(=)h Fu(f)p Fl(mk)d Fv(+)d(1)p Fl(;)17 b(:)g(:)g(:)33 b(;)17 b(mk)25 b Fv(+)d Fl(k)s Fu(g)32 b Fv(for)g(some)h(0)27 b Fu(\024)h Fl(m)g(<)g Fu(b)3677 2640 y Fp(n)p 3677 2656 43 4 v 3679 2714 a(k)3730 2679 y Fu(c)959 2939 y Fv(0)97 b(otherwise)330 3205 y(and)33 b(w)m(e)g(enco)s(de)h(the)f(w)m(eigh)m(ts)g Fl(w)1571 3220 y Fp(i)1632 3205 y Fv(of)f(the)h(edges)g Fl(e)2216 3220 y Fp(i)2277 3205 y Fv(in)f Fl(B)h Fv(=)27 b(\()p Fl(b)2680 3220 y Fp(i)2704 3229 y Fg(1)2739 3220 y Fp(i)2763 3229 y Fg(2)2798 3220 y Fp(:::)o(i)2881 3232 y Fe(k)2923 3205 y Fv(\):)857 3630 y Fl(b)898 3645 y Fp(i)922 3654 y Fg(1)957 3645 y Fp(i)981 3654 y Fg(2)1016 3645 y Fp(:::i)1100 3657 y Fe(k)1169 3630 y Fv(=)1272 3336 y Fj(8)1272 3426 y(>)1272 3456 y(>)1272 3485 y(>)1272 3515 y(<)1272 3695 y(>)1272 3725 y(>)1272 3754 y(>)1272 3784 y(:)1361 3527 y Fl(w)1431 3542 y Fp(e)1468 3527 y Fl(=)1517 3446 y Fj(\000)1562 3483 y Fp(n)1564 3561 y(k)1605 3446 y Fj(\001)1748 3527 y Fv(if)64 b Fu(f)p Fl(i)1953 3542 y Fo(1)1992 3527 y Fl(;)17 b(:)g(:)g(:)f(i)2200 3542 y Fp(k)2243 3527 y Fu(g)28 b Fv(=)f(edge)33 b Fl(e)g Fv(for)f(some)h Fl(e)28 b Fu(2)g Fl(E)1361 3787 y Fv(0)338 b(otherwise)330 4053 y(Then)34 b(this)e(problem)f(reduces)j(to)e(maximizing)d(Equation)j (\(4.1\))g(for)f(the)i(tensor)g Fl(C)39 b Fv(giv)m(en)33 b(b)m(y:)1605 4365 y Fl(c)1647 4319 y Fp(i)1671 4328 y Fg(1)1706 4319 y Fp(i)1730 4328 y Fg(2)1764 4319 y Fp(:::)o(i)1847 4331 y Fe(k)1647 4391 y Fp(j)1676 4400 y Fg(1)1710 4391 y Fp(j)1739 4400 y Fg(2)1773 4391 y Fp(:::)o(j)1861 4403 y Fe(k)1931 4365 y Fv(=)27 b Fl(a)2085 4380 y Fp(i)2109 4389 y Fg(1)2144 4380 y Fp(i)2168 4389 y Fg(2)2203 4380 y Fp(:::)n(i)2285 4392 y Fe(k)2328 4365 y Fl(b)2369 4380 y Fp(j)2398 4389 y Fg(1)2432 4380 y Fp(j)2461 4389 y Fg(2)2495 4380 y Fp(:::)o(j)2583 4392 y Fe(k)330 4631 y Ft(4.2)112 b(Notes)37 b(on)h(Heuristics)471 4946 y Fv(As)48 b(men)m(tioned)g(in)f(Section)g(1.3.2,)k(in)c(ligh)m(t) f(of)h(the)h(practical)e(v)-5 b(alue)47 b(and)h(theoretical)330 5163 y(di\016cult)m(y)g(of)f(the)h Fk(QAP)p Fv(,)h(there)g(has)f(b)s (een)h(an)f(e\013ort)f(to)h(\014nd)g(e\013ectiv)m(e)h(heuristics)f(for) f(the)330 5380 y(problem.)i(There)36 b(is)f(a)f(go)s(o)s(d)g(surv)m(ey) j(of)e(sev)m(eral)g(heuristics)g(in)f([BC)-56 b(\030)14 b(PP99].)51 b(In)35 b(this)g(section,)p eop %%Page: 82 90 82 89 bop 2070 -210 a Fw(82)330 91 y Fv(w)m(e)39 b(consider)f(our)g (results)h(from)d(the)j(p)s(ersp)s(ectiv)m(e)g(of)f(designing)f(and)h (analyzing)f(heuristics.)330 308 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Fv(and)37 b Fk(QAP)p Fv(,)i(the)e(branc)m(hing)h(is)f(t)m (ypically)f(done)h(b)m(y)i(\014xing)e(certain)g(parts)g(of)g(the)h (assignmen)m(t.)330 1391 y(The)31 b(idea)f(is)f(that)h(if)f(the)h (\014xed)i(part)e(of)f(the)i(assignmen)m(t)f(is)f(wrong,)i(then)g(the)f (branc)m(h)h(can)g(b)s(e)330 1608 y(cut.)471 1825 y(Giv)m(en)43 b(a)h(bullsey)m(e-t)m(yp)s(e)g(distribution,)g(this)f(approac)m(h)h (can)g(b)s(e)f(quite)h(successful.)77 b(Af-)330 2042 y(ter)35 b(\014nding)f(some)h(p)s(erm)m(utations)f(with)g(large)f(ob)5 b(jectiv)m(e)36 b(v)-5 b(alues)35 b(\(ideally)d(the)k(optim)m(um\),)d (it)330 2258 y(should)k(b)s(e)g(p)s(ossible)f(to)h(cut)g(man)m(y)g (branc)m(hes)i(that)d(lie)g(in)g(regions)g(far)h(from)e(the)j(optim)m (um,)330 2475 y(and)33 b(hence)h(ha)m(v)m(e)g(a)e(lo)m(w)g(a)m(v)m (erage)i(v)-5 b(alue.)471 2692 y(In)49 b(the)g(situation)d(of)i(a)g (spik)m(e-t)m(yp)s(e)i(distribution,)h(it)c(seems)i(that)f(it)g(should) g(b)s(e)g(v)m(ery)330 2908 y(di\016cult)31 b(to)g(implemen)m(t)f(an)h (e\013ectiv)m(e)i(branc)m(h-and-b)s(ound)f(pro)s(cedure,)h(since)f(the) g(n)m(umerous)330 3125 y(large)g(v)-5 b(alues)34 b(far)f(from)f(the)i (maxim)m(um)e(will)f(app)s(ear)i(in)g(almost)f(ev)m(ery)k(branc)m(h.)47 b(This)34 b(lea)m(v)m(es)330 3342 y(little)c(ro)s(om)h(for)h(error)h (in)e(the)i(estimates)g(of)f(the)h(maxim)m(um)d(on)i(eac)m(h)i(branc)m (h.)471 3558 y(W)-8 b(e)38 b(remark)e(that)h(if)f(w)m(e)i(b)s(eliev)m (e)f(w)m(e)h(ha)m(v)m(e)g(disco)m(v)m(ered)h(the)e(optim)m(um,)f(but)i (lac)m(k)f(an)f(op-)330 3775 y(timalit)m(y)i(pro)s(of,)k(then,)i(after) c(shifting)g(this)g(p)s(oin)m(t)g(to)h(the)g(iden)m(tit)m(y)-8 b(,)43 b(w)m(e)f(can)f(calculate)e(the)330 3992 y(cen)m(tral)29 b(pro)5 b(jection)28 b(\(see)i(Section)f(3.9\),)g(and)g(hence)h(the)f (distribution)e(around)i(the)g(presumed)330 4208 y(optim)m(um.)82 b(W)-8 b(e)47 b(hop)s(e)f(that)g(this)g(t)m(yp)s(e)h(of)e(qualitativ)m (e)g(information)e(w)m(ould)j(b)s(e)g(useful)h(in)330 4425 y(designing)32 b(branc)m(h-and-b)s(ound)h(algorithms.)330 4691 y Ft(4.3)112 b(Computational)36 b(Results)471 5007 y Fv(In)28 b(this)f(section,)i(w)m(e)f(apply)g(our)f(metho)s(ds)g(to)h (some)f(examples)g(that)h(are)f(considered)i(to)e(b)s(e)330 5223 y(of)36 b(practical)f(in)m(terest.)57 b(W)-8 b(e)37 b(compute)g(the)g(cen)m(tral)g(pro)5 b(jection)36 b(around)h(the)g (optim)m(um,)f(and)p eop %%Page: 86 94 86 93 bop 2070 -210 a Fw(86)330 91 y Fv(sample)32 b(p)s(oin)m(ts)g(to)g (gauge)h(ho)m(w)g(w)m(ell)f(this)g(appro)m(ximates)g(the)h (distribution.)471 308 y(W)-8 b(e)44 b(restrict)f(our)g(atten)m(tion)g (to)f(functions)h(of)g(t)m(yp)s(e)h(\(1.1\))f(for)g(t)m(w)m(o)g (reasons.)77 b(First,)45 b(as)330 525 y(w)m(e)d(remark)m(ed)f(in)g (Section)f(1.3,)j(the)e(generalized)g(problem)e(\(1.2\))i(is)f(seen)i (m)m(uc)m(h)g(less)f(often)330 741 y(in)d(practice.)62 b(Second,)41 b(it)d(follo)m(ws)g(from)f(Remark)h(3.2.5)h(that)f(an)m(y) i(function)e(in)g(the)h(cen)m(tral)330 958 y(cone)32 b(\(Section)f(3.3\))g(is)g(actually)f(an)h(ob)5 b(jectiv)m(e)32 b(function)f(of)g(t)m(yp)s(e)h(\(1.2\).)43 b(Then)32 b(for)f(ob)5 b(jectiv)m(e)330 1175 y(functions)46 b(of)f(this)h(form,)i (the)e(range)g(of)f(p)s(ossible)g(distributions)g(is)g(quite)h(wide,)j (and)d(our)330 1391 y(estimates)i(are)h(tigh)m(t.)92 b(F)-8 b(or)48 b(functions)g(of)h(t)m(yp)s(e)g(\(1.1\))g(it)e(is)i(p)s (ossible)f(that)g(the)i(range)e(of)330 1608 y(distributions)c(is)h (more)f(restricted,)49 b(and)d(that)f(our)g(estimates)g(could)g(b)s(e)g (impro)m(v)m(ed.)82 b(The)330 1825 y(computations)32 b(in)f(this)i(section)f(o\013er)h(some)f(data)g(to)h(fuel)f(sp)s (eculation.)471 2042 y(A)40 b(n)m(um)m(b)s(er)g(of)f(in)m(teresting)g (examples)h(of)f Fk(QAP)p Fv('s)i(of)e(t)m(yp)s(e)i(\(1.1\))e(ha)m(v)m (e)i(b)s(een)g(collected)e(in)330 2258 y(the)29 b Fk(QAPLIB)h Fv([BKR97)o(].)43 b(These)30 b(are)f(the)g(standard)g(test)g(sets)h (for)e(computational)e(approac)m(hes)330 2475 y(to)36 b(the)g Fk(QAP)p Fv(.)h(As)f(suc)m(h,)j(they)e(ha)m(v)m(e)g(the)f(adv) -5 b(an)m(tage)36 b(that)g(man)m(y)g(of)g(them)f(ha)m(v)m(e)j(b)s(een)e (solv)m(ed)330 2692 y(through)41 b(y)m(ears)i(of)e(e\013ort.)71 b(Unfortunately)-8 b(,)43 b(it)e(is)g(still)e(not)i(feasible)g(to)g (solv)m(e)h Fk(QAP)p Fv('s)h(larger)330 2908 y(than)34 b(ab)s(out)f Fl(n)d Fv(=)f(30)k(to)h(optimalit)m(y)-8 b(.)43 b(F)-8 b(or)33 b(larger)g(problems,)g([LP92])h(suggests)h(a)e (metho)s(d)g(of)330 3125 y(generating)e Fk(QAP)p Fv('s)h(that)f(ha)m(v) m(e)i(a)e(kno)m(wn)h(optimal)c(solution,)j(but)g(are)g(su\016cien)m (tly)h(generic)f(for)330 3342 y(use)38 b(in)e(testing)g(algorithms.)54 b(W)-8 b(e)37 b(include)f(some)h(results)g(on)g(these)h(generated)g (problems)e(in)330 3558 y(Section)c(4.3.3.)471 3775 y(W)-8 b(e)33 b(also)f(tak)m(e)i(examples)e(from)g(the)h(Nugen)m(t,)g(or)g Fk(nug)g Fv(series)h(of)e(problems)g(in)m(tro)s(duced)h(in)330 3992 y([NVR68].)71 b(This)41 b(series)i(is)e(w)m(ell)g(suited)h(for)f (our)h(exp)s(erimen)m(ts,)i(pro)m(viding)d(sev)m(eral)h(related)330 4208 y(problems)36 b(based)i(on)f(a)f(simple)g(structure.)57 b(These)39 b(problems)d(ha)m(v)m(e)i(b)s(een)g(among)d(the)i(most)330 4425 y(studied)48 b Fk(QAP)g Fv(test)g(cases,)53 b(and)48 b(ha)m(v)m(e)g(no)m(w)h(b)s(een)f(solv)m(ed)g([ABGL02)o(].)89 b(They)49 b(are)e(deriv)m(ed)330 4642 y(from)28 b(the)i(t)m(yp)s(e)h (of)e(\\facilit)m(y)e(lo)s(cation")h(problem)g(suggested)j(b)m(y)g(Ko)s (opmans-Bec)m(kmann)e(\(see)330 4858 y(Section)46 b(1.3\).)82 b(The)47 b(problems)e(in)m(v)m(olv)m(e)h(placing)e(factories)h(on)h(a)f (giv)m(en)h(rectangular)f(grid)330 5075 y(\(sa)m(y)39 b(5)26 b Fu(\002)h Fv(6)38 b(for)f(n=30\))h(with)g(Manhattan)g(\()p Fl(L)2110 5039 y Fo(1)2150 5075 y Fv(\))g(distances.)62 b(So)38 b(the)h(distance)f(matrix)f Fl(A)h Fv(is)330 5292 y(symmetric,)f(with)f(zero)h(diagonal,)f(and)h(small)d(p)s(ositiv) m(e)i(in)m(teger)h(en)m(tries)g(\(at)f(most)g(9)h(in)e(the)p eop %%Page: 87 95 87 94 bop 2070 -210 a Fw(87)330 91 y Fv(5)7 b Fu(\002)g Fv(6)24 b(case\).)42 b(The)26 b(\015o)m(w)g(matrix)d Fl(B)31 b Fv(is)24 b(also)g(symmetric)g(and)i(consists)f(of)g(small)e (p)s(ositiv)m(e)h(in)m(teger)330 308 y(en)m(tries.)53 b(The)36 b(original)c(examples)k(w)m(ere)h(for)e Fl(n)d Fv(=)h(12)p Fl(;)17 b Fv(15)p Fl(;)g Fv(20)p Fl(;)g Fv(30.)49 b(The)37 b(remaining)c(examples)330 525 y(w)m(ere)j(created)g(later)d (b)m(y)j(deleting)e(ro)m(ws)h(and)g(columns)f(\(factories\))h(from)e (the)i(\015o)m(w)h(matrices,)330 741 y(and)d(building)d(the)j(distance) g(matrices)f(from)f(new)j(\(sometimes)d(non-rectangular\))h(grids.)471 958 y(W)-8 b(e)29 b(classify)e(the)i Fk(nug)g Fv(problems)e(as)i(part)f (of)f(the)i(symmetric)e(sp)s(ecial)g(case)i(of)f(Section)g(2.3,)330 1175 y(but,)52 b(since)47 b(the)h(ro)m(w)g(and)g(columns)f(sums)h(v)-5 b(ary)d(,)51 b(not)c(part)g(of)g(the)h(pure)g(sp)s(ecial)f(case)h(of) 330 1391 y(Section)32 b(2.2.)330 1644 y Fs(4.3.1)105 b(Cen)m(tral)34 b(Pro)6 b(jections)37 b(for)e(the)f Fq(nug)g Fs(Problems)471 1925 y Fv(T)-8 b(o)32 b(compute)g(the)g(cen)m(tral)f (pro)5 b(jection,)32 b(w)m(e)h(use)f(the)g(metho)s(ds)g(of)f(Section)h (3.9.)43 b(W)-8 b(e)32 b(b)s(egin)330 2141 y(b)m(y)j(\014nding)f(the)g (global)e(minim)m(um)f Fl(\034)11 b Fv(,)35 b(relying)e(on)h(the)h(w)m (ork)g(compiled)d(in)h(the)i Fk(QAPLIB)p Fv(,)h(and)330 2358 y(w)m(e)43 b(shift)f(the)h(problem)f(so)g(that)g Fl(\034)54 b Fv(lies)42 b(at)g(the)h(iden)m(tit)m(y)f Fl(")g Fv(\(see)h(Remark)g(3.1.2\).)72 b(W)-8 b(e)43 b(are)330 2575 y(in)m(terested)c(in)d(\014nding)i(out)f(whether)i(the)f (shap)s(e)g(of)f(the)h(cen)m(tral)g(pro)5 b(jection)37 b(of)g(the)h(shifted)330 2792 y(problem)28 b(is)h(a)g(bullsey)m(e)h (distribution)e(\(Section)h(2.1\),)h(a)f(spik)m(e)h(distribution)d (\(Section)j(2.3\),)f(or)330 3008 y(a)j(diluted)g(spik)m(e)i(with)e (prop)s(erties)h(of)f(b)s(oth)h(\(Section)f(4.2\).)44 b(F)-8 b(or)32 b(this)g(purp)s(ose,)i(w)m(e)f(translate)330 3225 y(the)40 b(ob)5 b(jectiv)m(e)41 b(function)e Fl(f)50 b Fv(so)40 b(that)g(it)f(has)h(an)f(a)m(v)m(erage)i(o)m(v)m(er)g(all)c (p)s(erm)m(utations)i(of)g(0,)j(and)330 3442 y(scale)33 b Fl(f)43 b Fv(so)33 b(that)f(it)f(has)i(a)g(maxim)m(um)d(\(rather)j (than)f(a)g(minim)m(um\))e(of)i(1)g(at)g(the)h(iden)m(tit)m(y)-8 b(.)471 3658 y(It)40 b(is)f(con)m(v)m(enien)m(t)j(to)d(w)m(ork)i(with)e (problems)g(that)h(are)f(symmetric)g(\(suc)m(h)j(as)d(the)i Fk(nug)p Fv('s\))330 3875 y(and)33 b(ha)m(v)m(e)g(ev)m(en)h Fl(n)p Fv(.)44 b(Then,)34 b(after)e(prepro)s(cessing,)h(the)g(cen)m (tral)f(pro)5 b(jection)32 b Fl(g)k Fv(of)c Fl(f)43 b Fv(m)m(ust)33 b(b)s(e)f(a)330 4092 y(con)m(v)m(ex)k(com)m(bination)31 b(of)i(the)i(extreme)f(functions)f Fl(r)2326 4107 y Fo(3)2399 4092 y Fv(and)h Fl(r)2634 4107 y Fo(7)p Fp(e)2740 4092 y Fv(generating)f(the)h(ev)m(en)h(cen)m(tral)330 4308 y(symmetric)d(cone)h Fl(K)1110 4323 y Fp(s)1180 4308 y Fv(\(see)h(Section)f(3.7\).)43 b(That)34 b(is,)e Fl(g)f Fv(=)d Fl(\013)2565 4323 y Fo(1)2605 4308 y Fl(r)2649 4323 y Fo(3)2711 4308 y Fv(+)22 b Fl(\013)2871 4323 y Fo(2)2910 4308 y Fl(r)2954 4323 y Fo(7)p Fp(e)3026 4308 y Fv(,)33 b(where)h Fl(\013)3430 4323 y Fo(1)3492 4308 y Fv(+)22 b Fl(\013)3652 4323 y Fo(2)3720 4308 y Fv(=)28 b(1.)330 4525 y(So)41 b(w)m(e)h(need)g(to)f(\014nd)h Fl(\013)1256 4540 y Fo(1)1295 4525 y Fv(.)70 b(Ha)m(ving)41 b(\014xed)h(the)f(v)-5 b(alue)41 b(at)g(the)g(iden)m(tit)m(y)g Fl(")g Fv(to)g(b)s(e)g(1,)i(w)m(e)g(note)330 4742 y(that)27 b Fl(\013)598 4757 y Fo(1)665 4742 y Fv(is)g(completely)g(determined)g (b)m(y)h(the)g(v)-5 b(alue)27 b(of)g Fl(g)k Fv(on)c(an)m(y)h(conjugacy) h(class)e(other)h(than)330 4958 y(the)36 b(class)g(con)m(taining)e(the) i(iden)m(tit)m(y)-8 b(.)52 b(The)36 b(smallest)e(suc)m(h)j(class)f(is)f (the)h(set)g(of)f(2-cycles.)52 b(W)-8 b(e)330 5175 y(observ)m(e)29 b(that)f Fl(r)923 5190 y Fo(3)990 5175 y Fv(has)g(an)f(a)m(v)m(erage)i (v)-5 b(alue)27 b(of)g(1)g(on)g(2-cycles,)i(while)e Fl(r)2871 5190 y Fo(7)p Fp(e)2971 5175 y 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Fl(n)330 4265 y Fv(increases.)471 4481 y(W)-8 b(e)40 b(ha)m(v)m(e)g(computed)g (the)f(standard)h(deviation)e(of)h(the)h(appro)m(ximation)d(ratios)h(o) m(v)m(er)i(all)330 4698 y(p)s(erm)m(utations)25 b(in)g(T)-8 b(able)25 b(4.4.)41 b(This)26 b(is)f(obtained)g(b)m(y)i(dividing)c(the) k(standard)f(deviation)e(of)h(the)p 465 4820 3300 4 v 463 4933 4 113 v 480 4933 V 651 4899 a Fr(n)g Fw(=)p 969 4933 V 986 4933 V 294 w(12)p 1293 4933 V 216 w(14)p 1599 4933 V 217 w(16)p 1906 4933 V 217 w(18)p 2213 4933 V 216 w(20)p 2519 4933 V 217 w(22)p 2826 4933 V 217 w(24)p 3133 4933 V 216 w(28)p 3440 4933 V 217 w(30)p 3746 4933 V 3763 4933 V 465 4936 3300 4 v 463 5049 4 113 v 480 5049 V 532 5016 a Fr(S)5 b(td:)31 b(D)s(ev)s(:)p 969 5049 V 986 5049 V 116 w Fw(0.213)p 1293 5049 V 102 w(0.188)p 1599 5049 V 102 w(0.170)p 1906 5049 V 102 w(0.156)p 2213 5049 V 101 w(0.143)p 2519 5049 V 102 w(0.151)p 2826 5049 V 102 w(0.125)p 3133 5049 V 101 w(0.115)p 3440 5049 V 102 w(0.109)p 3746 5049 V 3763 5049 V 465 5053 3300 4 v 566 5307 a(T)-8 b(able)30 b(4.3:)42 b(Standard)29 b(Deviation)h(of)h (Appro)m(ximation)e(Ratios)h(for)g(the)h Fq(nug)f Fw(Problems)p eop %%Page: 91 99 91 98 bop 2070 -210 a Fw(91)650 2288 y @beginspecial 63 @llx 210 @lly 550 @urx 589 @ury 3515 @rwi @setspecial %%BeginDocument: nug30_hist.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/math.lsa.umich.edu/group/grad/tamon/QAP/Numeriment/Thesis_trials/nug30.eps %%CreationDate: 03/30/ 2 12:40:12 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 63 210 550 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 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/FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub 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b(of)f(the)h(ob)5 b(jectiv)m(e)24 b(function)f(o)m(v)m(er)h(all)e Fl(\033)27 b Fv(that)c(restrict)g(to)g Fl(\033)2811 4755 y Fp(S)2886 4740 y Fv(in)f(their)h(\014rst)h Fl(S)29 b Fv(p)s(ositions:)1652 5005 y Fl(E)22 b Fv([)p Fl(\033)t Fu(j)p Fl(\033)1915 5020 y Fp(S)1966 5005 y Fv(])28 b(=)2174 4911 y Fj(X)2159 5131 y Fp(\033)r Ff(2)p Fp(S)2291 5139 y Fe(n)2125 5217 y Fp(\033)2165 5231 y Fh(j)p Fe(S)2228 5217 y Fo(=)p Fp(\033)2323 5228 y Fe(S)2385 5005 y Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))330 5380 y(Using)29 b(the)h(observ)-5 b(ations)30 b(of)f(Lemma)f(3.1.1,)i(w)m(e)h(can)f (compute)g(the)g(conditional)d(exp)s(ectation)p eop %%Page: 96 104 96 103 bop 2070 -210 a Fw(96)330 91 y Fv(in)36 b(time)g Fl(O)s Fv(\()p Fl(n)849 55 y Fo(2)888 91 y Fv(\))h(for)f(ob)5 b(jectiv)m(e)38 b(functions)f(of)g(t)m(yp)s(e)h(\(1.1\))e(and)i(in)e (time)g Fl(O)s Fv(\()p Fl(n)3251 55 y Fo(4)3289 91 y Fv(\))h(for)g(ob)5 b(jectiv)m(e)330 308 y(function)45 b(of)g(t)m(yp)s(e)h(\(1.2\).)82 b(This)45 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(h)i(agrees)f(with)g Fl(\034)42 b Fv(on)31 b Fl(S)2804 4043 y Fp(n)2851 4028 y Fv(.)43 b(Th)m(us)33 b(w)m(e)f(can)f(get)g(a)g (par-)330 4245 y(tial)36 b(assignmen)m(t)h Fl(\033)1071 4260 y Fp(S)1114 4269 y Fg(0)1191 4245 y Fv(that)g(has)h(conditional)e (exp)s(ectation)h(at)h(least)f Fl(\013)q(=n)3154 4209 y Fo(2)3231 4245 y Fv(b)m(y)i(en)m(umerating)330 4462 y(the)33 b Fl(n)556 4477 y Fo(\()p Fp(k)r Fo(\))686 4462 y Fv(p)s(ossible)f(assignmen)m(ts)h(of)f(the)h Fl(n)1931 4477 y Fo(\()p Fp(k)r Fo(\))2061 4462 y Fl(k)s Fv(-subsets)i(of)d Fu(f)p Fv(1)p Fl(;)17 b Fv(2)p Fl(;)g(:)g(:)g(:)e(n)p Fu(g)p Fv(.)471 4678 y(No)m(w)42 b(w)m(e)g(will)d(extend)k(the)f (partial)d(assignmen)m(t)i Fl(\033)2439 4693 y Fp(S)2482 4702 y Fg(0)2562 4678 y Fv(in)m(to)f(a)h(p)s(erm)m(utation)f Fl(\033)45 b Fv(whic)m(h)d(at)330 4895 y(least)35 b(meets)h(the)g(exp)s (ectation,)h(using)e(the)h(Greedy)h(Exp)s(ectation)e(Algorithm.)50 b(In)36 b(this)g(pro-)330 5112 y(cedure,)43 b(w)m(e)e(assign)e(the)i(v) -5 b(alues)39 b(of)h Fu(f)p Fv(1)p Fl(;)17 b Fv(2)p Fl(;)g(:)g(:)g(:)e (n)p Fu(g)27 b Fi(r)g Fl(S)2378 5127 y Fo(0)2457 5112 y Fv(sequen)m(tially)40 b(using)f(conditional)f(ex-)330 5328 y(p)s(ectation.)49 b(Let)34 b Fl(S)j Fu(\023)31 b Fl(S)1246 5343 y Fo(0)1320 5328 y Fv(b)s(e)k(the)g(set)g(of)f(curren) m(tly)h(assigned)g(v)-5 b(alues,)35 b(and)g Fl(\033)3266 5343 y Fp(S)3351 5328 y Fv(b)s(e)g(the)g(corre-)p eop %%Page: 97 105 97 104 bop 2070 -210 a Fw(97)330 91 y Fv(sp)s(onding)35 b(partial)e(assignmen)m(t.)51 b(If)35 b Fl(l)i Fv(is)e(the)g(lo)m(w)m (est)h(remaining)d(unassigned)j(v)-5 b(alue)34 b(\(that)h(is,)330 308 y(min)15 b Fu(ff)p Fv(1)p Fl(;)i Fv(2)p Fl(;)g(:)g(:)g(:)e(n)p Fu(g)22 b Fi(r)h Fl(S)6 b Fu(g)o Fv(\),)41 b(w)m(e)f(compute)f(the)h (conditional)c(exp)s(ectations)k(for)f(eac)m(h)h(p)s(ossible)330 525 y(assignmen)m(t)31 b(of)g Fl(\033)t Fv(\()p Fl(l)r Fv(\).)44 b(A)m(t)32 b(least)f(one)h(of)f(these)i(m)m(ust)e(meet)h(the) g(o)m(v)m(erall)e(exp)s(ectation)i(of)f Fl(\033)t Fv(\()p Fl(S)6 b Fv(\).)330 741 y(W)-8 b(e)33 b(c)m(ho)s(ose)g(suc)m(h)h(a)f(v) -5 b(alue)32 b(\(sa)m(y)-8 b(,)33 b(the)g(largest\),)f(and)h(\014x)g (this)f(to)g(b)s(e)h(the)g(v)-5 b(alue)32 b(of)g Fl(\033)t Fv(\()p Fl(l)r Fv(\).)471 958 y(By)g(iterating)d(this)i(pro)s(cedure,)h (w)m(e)g(build)e(a)h(p)s(erm)m(utation)e(with)i(exp)s(ected)i(v)-5 b(alue)31 b(at)f(least)330 1175 y Fl(\013)q(=n)500 1139 y Fo(2)539 1175 y Fv(.)42 b(The)30 b(total)e(time)g(for)g(this)h(pro)s (cedure)h(is)f Fl(O)s Fv(\()p Fl(n)2304 1190 y Fo(\()p Fp(k)r Fo(\))2416 1175 y Fu(\001)15 b Fl(n)2517 1190 y Fo(\()p Fp(k)r Fo(\))2630 1175 y Fu(\001)g Fl(n)2731 1139 y Fo(2)2785 1175 y Fv(+)g Fl(n)2934 1139 y Fo(2)2989 1175 y Fu(\001)g Fl(n)g Fu(\001)g Fl(n)p Fv(\))27 b(=)h Fl(O)s Fv(\()p Fl(n)3549 1139 y Fo(2)p Fp(k)r Fo(+2)3716 1175 y Fv(\))h(for)330 1391 y(ob)5 b(jectiv)m(e)34 b(functions)e(of)g (t)m(yp)s(e)i(\(1.1\),)e(and)h Fl(O)s Fv(\()p Fl(n)2120 1355 y Fo(2)p Fp(k)r Fo(+4)2287 1391 y Fv(\))g(for)f(ob)5 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y Fp(ii)3330 4494 y(i)3354 4475 y Fh(0)3376 4494 y Fp(i)3400 4475 y Fh(0)1212 4830 y Fv(+)1668 4763 y(1)p 1298 4807 791 4 v 1298 4898 a(\()p Fl(n)22 b Fu(\000)g Fl(k)s Fv(\)\()p Fl(n)h Fu(\000)f Fl(k)j Fu(\000)e Fv(1\))2127 4735 y Fj(X)2114 4960 y Fp(i;j)11 b(=)-42 b Ff(2)o Fp(S)2144 5051 y(i)p Ff(6)p Fo(=)p Fp(j)2455 4735 y Fj(X)2350 4961 y Fp(i)2374 4937 y Fh(0)2396 4961 y Fp(;j)2449 4937 y Fh(0)2478 4961 y Fp(=)g Ff(2)p Fp(\033)2558 4972 y Fe(S)2602 4961 y Fo(\()p Fp(S)t Fo(\))2449 5056 y Fp(i)2473 5037 y Fh(0)2495 5056 y Ff(6)p Fo(=)p Fp(j)2583 5037 y Fh(0)2721 4830 y Fl(c)2763 4783 y Fp(ij)2763 4858 y(i)2787 4839 y Fh(0)2809 4858 y Fp(j)2842 4839 y Fh(0)p eop %%Page: 98 106 98 105 bop 2070 -210 a Fw(98)330 91 y Ft(4.5)112 b(F)-9 b(urther)38 b(Questions)471 407 y Fv(The)g(estimates)f(of)g(Theorems)h (2.1.1,)f(2.2.1,)g(2.3.1)g(and)g(2.4.1)g(for)g(the)h(n)m(um)m(b)s(er)f (of)g(near-)330 623 y(optimal)27 b(p)s(erm)m(utations)h(can)i(b)s(e)f (used)i(to)e(b)s(ound)h(the)f(optimal)e(v)-5 b(alue)29 b(b)m(y)h(a)f(sample)g(optim)m(um)330 840 y(in)k(branc)m(h-and-b)s (ound)h(algorithms.)43 b(Those)35 b(estimates)e(are)g(\(nearly\))h(b)s (est)g(p)s(ossible)f(for)f(the)330 1057 y(generalized)f(problem)e (\(1.2\).)43 b(Ho)m(w)m(ev)m(er,)34 b(it)c(is)g(not)h(clear)g(whether)h (they)g(can)f(b)s(e)h(impro)m(v)m(ed)f(in)330 1273 y(the)36 b(case)h(of)e(standard)h Fk(QAP)h Fv(\(1.1\))e(or)h(ho)m(w)g(to)f (impro)m(v)m(e)h(them)f(in)g(in)m(teresting)h(sp)s(ecial)e(cases.)330 1490 y(In)f(particular,)e(w)m(e)i(ask)h(the)f(follo)m(wing)d(question:) 462 1781 y Fu(\017)45 b Fv(Let)33 b Fl(f)39 b Fv(:)28 b Fl(S)934 1796 y Fp(n)1009 1781 y Fu(\000)-16 b(!)28 b Fi(R)44 b Fv(b)s(e)33 b(the)g(ob)5 b(jectiv)m(e)34 b(function)e(in)g(the)i(T)-8 b(ra)m(v)m(eling)32 b(Salesman)g(Problem) 557 1998 y(\(cf.)49 b(Section)34 b(1.2\),)g(let)p 1454 1917 59 4 v 34 w Fl(f)44 b Fv(b)s(e)35 b(the)f(a)m(v)m(erage)i(v)-5 b(alue)33 b(of)h Fl(f)45 b Fv(and)34 b(let)g Fl(f)3043 2013 y Fo(0)3113 1998 y Fv(=)c Fl(f)k Fu(\000)p 3402 1917 V 24 w Fl(f)10 b Fv(.)49 b(Let)34 b Fl(\034)46 b Fv(b)s(e)557 2215 y(an)37 b(optimal)e(p)s(erm)m(utation,)i(so)g(that)g Fl(f)2039 2230 y Fo(0)2079 2215 y Fv(\()p Fl(\034)11 b Fv(\))36 b Fu(\025)g Fl(f)2405 2230 y Fo(0)2444 2215 y Fv(\()p Fl(\033)t Fv(\))h(for)g(all)e Fl(\033)k Fu(2)d Fl(S)3166 2230 y Fp(n)3213 2215 y Fv(.)58 b(Is)37 b(it)g(true)g(that) 557 2431 y(for)f(an)m(y)h(\014xed)h Fl(\015)h(>)34 b Fv(0)j(there)g(is)f(a)g(n)m(um)m(b)s(er)h Fl(\016)h Fv(=)c Fl(\016)t Fv(\()p Fl(\015)5 b Fv(\))34 b Fl(>)h Fv(0)h(suc)m(h)i(that)e (the)h(probabilit)m(y)557 2648 y(that)29 b(a)g(random)g(p)s(erm)m (utation)e Fl(\033)32 b Fu(2)c Fl(S)1993 2663 y Fp(n)2069 2648 y Fv(satis\014es)i(the)g(inequalit)m(y)e Fl(f)3087 2663 y Fo(0)3127 2648 y Fv(\()p Fl(\033)t Fv(\))f Fu(\025)3405 2581 y Fl(\015)p 3404 2625 V 3404 2716 a(n)3472 2648 y(f)3520 2663 y Fo(0)3560 2648 y Fv(\()p Fl(\034)11 b Fv(\))30 b(is)e(at)557 2865 y(least)k Fl(n)843 2829 y Ff(\000)p Fp(\016)969 2865 y Fv(for)g(all)e(su\016cien)m(tly)k(large)d Fl(n)p Fv(?)330 3156 y(The)c(small)c(sample)i(of)h(results)g(in)f (Section)h(4.3)f(suggests)i(that)f(the)g(answ)m(er)i(is)d(\\y)m(es".)42 b(One)27 b(w)m(a)m(y)330 3373 y(that)j(w)m(e)g(could)g(try)g(to)f(tigh) m(ten)h(the)g(analysis)f(is)g(to)h(use)g(information)d(from)h(the)j (standard)f(de-)330 3590 y(viation)f(and)h(higher)g(momen)m(ts)g(of)g (the)h(distribution)d(whic)m(h)j(are)f(computable)g(in)f(p)s(olynomial) 330 3806 y(time.)471 4023 y(In)34 b([Bar],)f(there)h(is)f(a)g (preliminary)d(results)k(in)e(this)h(direction,)f(obtained)h(b)m(y)h (relating)d(the)330 4240 y Fl(L)396 4203 y Ff(1)504 4240 y Fv(norm)g(of)h(a)h(function)f(to)g(its)g Fl(L)1654 4203 y Fo(2)p Fp(k)1765 4240 y Fv(norm)g(\(that)g(is,)g(its)g(2)p Fl(k)s Fv(-th)g(momen)m(t\):)330 4531 y Ft(Theorem)37 b(4.5.1)h(\(Barvinok\).)145 b Fd(F)-8 b(or)36 b(an)m(y)h Fl(\013)e(>)f Fv(0)p Fd(,)j(there)g(exists)g(a)f Fl(\026)e Fv(=)g Fl(\026)p Fv(\()p Fl(\013)q Fv(\))f Fl(>)h Fv(0)i 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y(do)e(this)h(is)f(just)h(to)f(hard)h(co)s(de)f(a)h(large)e(n)m(um)m(b) s(er,)i(lik)m(e)f(10)2497 4223 y Fo(10)2571 4259 y Fv(.)330 4475 y Fk (\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045)p (\045\045\045\045\045\045\045\045)q(\045)57 b(relaxed_max.m)e (\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045)p (\045\045\045\045\045\045)330 4616 y(\045)330 4757 y(function)e(X)f(=)f (relaxed_max\(A,B\))330 4898 y(\045)330 5039 y(\045)h(Given)g (diagonalizable)j(matrices)f(A)d(and)h(B,)g(this)g(function)i(should) 330 5181 y(\045)e(return)g(orthogonal)i(matrix)f(X)f(maximizing)i (.)330 5322 y(\045)e(It)f(does)i(this)f(by)g(transforming)i (the)e(problem)h(to)f(the)g(diagonal)i(version,)p eop %%Page: 105 113 105 112 bop 2047 -210 a Fw(105)330 91 y Fk(\045)52 b(and)g(then)g (finding)h(a)f(permutation)i(which)e(puts)h(the)f(elements)h(in)f(the)g (same)330 232 y(\045)g(order.)104 b(Because)53 b(diagonalizing)i(uses)d (orthogonal)i(matrices,)f(we)f(can)330 374 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(tral)g(pro)5 b(jection)32 b(of)g(the)h(distribution)e(around)h(the)h (iden)m(tit)m(y)-8 b(.)p eop %%Page: 106 114 106 113 bop 2047 -210 a Fw(106)330 91 y Fk (\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045)p (\045\045\045\045\045\045\045\045)q(\045)57 b(sum_on_2cy.m)e (\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045)p (\045\045\045\045\045\045)330 232 y(\045)330 374 y(function)e(Z)f(=)f (sum_on_2cy\(M,A,B\))330 515 y(\045)330 656 y(\045)h(Sums)g(the)g (values)h(of)f(tr\(X*A*X'*B'\))i(over)f(2-cycles)g(\(transpositions\))j (X.)330 797 y(\045)c(Inputs)g(A)g(and)g(B)g(are)g(M)f(by)h(M)f (matrices,)j(the)e(X's)g(are)g(matrix)330 938 y(\045)103 b(representations)55 b(of)d(a)f(permutation.)330 1079 y(\045)330 1220 y(Z)h(=)f(0;)923 b(\045)52 b(Sum)g(of)g(values)h(of)e (f)h(on)g(2-cycles)330 1362 y(for)g(i=1:M)433 1503 y(for)g(j=i+1:M)535 1644 y(Tr=ones\(1,M\);)824 b(\045)51 b(This)i(2-cycle.)535 1785 y(for)f(k=2:M)638 1926 y(Tr\(1,k\)=k;)874 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y(if)g(\(l)g(~=i)g(&)f(l~=j)i(&)e(l~=k\))1048 2632 y(pa1111)i(=)e(pa1111)i(+)f(A\(i,j\)*A\(k,l\);)1048 2773 y(pb1111)h(=)e(pb1111)i(+)f(B\(i,j\)*B\(k,l\);)945 2914 y(end)843 3055 y(end)843 3196 y(pa211b)h(=)e(pa211b)i(+)e (A\(i,j\)*A\(i,k\);)843 3337 y(pa211c)i(=)e(pa211c)i(+)e (A\(i,j\)*A\(k,i\);)843 3479 y(pa211d)i(=)e(pa211d)i(+)e (A\(j,i\)*A\(i,k\);)843 3620 y(pa211e)i(=)e(pa211e)i(+)e (A\(j,i\)*A\(k,i\);)843 3761 y(pb211b)i(=)e(pb211b)i(+)e (B\(i,j\)*B\(i,k\);)843 3902 y(pb211c)i(=)e(pb211c)i(+)e (B\(i,j\)*B\(k,i\);)843 4043 y(pb211d)i(=)e(pb211d)i(+)e (B\(j,i\)*B\(i,k\);)843 4184 y(pb211e)i(=)e(pb211e)i(+)e (B\(j,i\)*B\(k,i\);)740 4325 y(end)638 4467 y(end)638 4608 y(pa22b)h(=)g(pa22b)g(+)g(A\(i,j\)*A\(j,i\);)638 4749 y(pa22c)g(=)g(pa22c)g(+)g(A\(i,j\)*A\(i,j\);)638 4890 y(pb22b)g(=)g(pb22b)g(+)g(B\(i,j\)*B\(j,i\);)638 5031 y(pb22c)g(=)g(pb22c)g(+)g(B\(i,j\)*B\(i,j\);)535 5172 y(end)433 5313 y(end)p eop %%Page: 108 116 108 115 bop 2047 -210 a Fw(108)433 91 y Fk(pa4)52 b(=)f(pa4)h(+)g (A\(i,i\)*A\(i,i\);)433 232 y(pb4)g(=)f(pb4)h(+)g(B\(i,i\)*B\(i,i\);) 330 374 y(end)330 515 y(s1)g(=)f(pa1111*pb1111/M/\(M-1\)/\(M)q(-2\))q (/\(M-)q(3\);)330 656 y(s2)h(=)f(\(pa211b*pb211b+pa211c*pb)q(211)q (c+pa)q(211d)q(*pb2)q(11d)q(+pa2)q(11e*)q(pb21)q(1e\))q(/M/\()q(M-1\))q (/\(M-)q(2\);)330 797 y(s3)h(=)f(\(pa22b*pb22b+pa22c*pb22c)q(\)/M)q (/\(M-)q(1\);)330 938 y(s4)h(=)f(pa4*pb4;)330 1079 y(cvar)h(=)g (s1+s2+s3+s4;)471 1295 y Fv(Finally)25 b(w)m(e)k(giv)m(e)f(a)f(sample)g (input)g(\014le,)i Fk(solved)p 2278 1295 31 4 v 38 w(prob)p 2520 1295 V 38 w(data.m)p Fv(.)44 b(This)27 b(is)h(the)g(data)f(for)g (the)330 1436 y Fk(nug12)34 b Fv(problem.)330 1652 y Fk(function)53 b([M,A,B,P])h(=)e(solved_prob_data)330 1793 y (\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045)p (\045\045\045\045\045\045\045\045)q(\045)57 b(solved_prob_data_nug12.m) h(\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045\045)330 1934 y(\045)330 2076 y(\045)52 b(Contains)h(the)f(data)g(for)g(an)g (instance)i(of)d(QAP)h(to)g(try.)330 2217 y(\045)g(M)f(is)h(the)g (dimension)i(of)d(the)h(problem,)i(A)d(and)h(B)g(are)g(the)g(input)h (matrices.)330 2358 y(\045)f(Want)g(to)g(minimize)h(trace\(Z*A*Z'*B'\)) j(where)c(Z)g(is)g(a)f(permutation)j(matrix.)330 2499 y(\045)e(P)f(is)h(the)g(optimal)h(permutation,)i(found)d(by)g(years)h (of)e(hard)i(work)f(by)330 2640 y(\045)g(many)g(researchers.)330 2781 y(\045)330 2922 y(\045)g(This)g(data)g(is)g(known)h(as)e("nug12") 330 3064 y(\045)330 3205 y(M=12;)330 3346 y(A=[0)h(1)g(2)f(3)h(1)f(2)h (3)g(4)f(2)h(3)f(4)h(5;)330 3487 y(1)g(0)f(1)h(2)f(2)h(1)f(2)h(3)f(3)h (2)f(3)h(4;)330 3628 y(2)g(1)f(0)h(1)f(3)h(2)f(1)h(2)f(4)h(3)f(2)h(3;) 330 3769 y(3)g(2)f(1)h(0)f(4)h(3)f(2)h(1)f(5)h(4)f(3)h(2;)330 3910 y(1)g(2)f(3)h(4)f(0)h(1)f(2)h(3)f(1)h(2)f(3)h(4;)330 4051 y(2)g(1)f(2)h(3)f(1)h(0)f(1)h(2)f(2)h(1)f(2)h(3;)330 4193 y(3)g(2)f(1)h(2)f(2)h(1)f(0)h(1)f(3)h(2)f(1)h(2;)330 4334 y(4)g(3)f(2)h(1)f(3)h(2)f(1)h(0)f(4)h(3)f(2)h(1;)330 4475 y(2)g(3)f(4)h(5)f(1)h(2)f(3)h(4)f(0)h(1)f(2)h(3;)330 4616 y(3)g(2)f(3)h(4)f(2)h(1)f(2)h(3)f(1)h(0)f(1)h(2;)330 4757 y(4)g(3)f(2)h(3)f(3)h(2)f(1)h(2)f(2)h(1)f(0)h(1;)330 4898 y(5)g(4)f(3)h(2)f(4)h(3)f(2)h(1)f(3)h(2)f(1)h(0];)330 5039 y(B=[0)104 b(5)e(2)h(4)g(1)g(0)g(0)f(6)h(2)g(1)g(1)g(1;)330 5181 y(5)g(0)g(3)f(0)h(2)g(2)g(2)g(0)f(4)h(5)g(0)g(0;)330 5322 y(2)g(3)g(0)f(0)h(0)g(0)g(0)g(5)f(5)h(2)g(2)g(2;)p eop %%Page: 109 117 109 116 bop 2047 -210 a Fw(109)330 91 y Fk(4)103 b(0)g(0)f(0)h(5)g(2)g (2)51 b(10)103 b(0)g(0)g(5)g(5;)330 232 y(1)g(2)g(0)f(5)h(0)52 b(10)103 b(0)g(0)f(0)h(5)g(1)g(1;)330 374 y(0)g(2)g(0)f(2)52 b(10)103 b(0)g(5)g(1)f(1)h(5)g(4)g(0;)330 515 y(0)g(2)g(0)f(2)h(0)g(5)g (0)51 b(10)103 b(5)g(2)g(3)g(3;)330 656 y(6)g(0)g(5)51 b(10)103 b(0)g(1)52 b(10)103 b(0)f(0)h(0)g(5)g(0;)330 797 y(2)g(4)g(5)f(0)h(0)g(1)g(5)g(0)f(0)h(0)52 b(10)g(10;)330 938 y(1)103 b(5)g(2)f(0)h(5)g(5)g(2)g(0)f(0)h(0)g(5)g(0;)330 1079 y(1)g(0)g(2)f(5)h(1)g(4)g(3)g(5)51 b(10)103 b(5)g(0)g(2;)330 1220 y(1)g(0)g(2)f(5)h(1)g(0)g(3)g(0)51 b(10)103 b(0)g(2)g(0];)330 1362 y(P=[12,7,9,3,4,8,11,1,5,6,)q(10,2)q(];)p eop %%Page: 110 118 110 117 bop 1665 2806 a Ft(BIBLIOGRAPHY)2047 5680 y Fw(110)p eop %%Page: 111 119 111 118 bop 2047 -210 a Fw(111)1665 532 y Ft(BIBLIOGRAPHY)330 1165 y Fw([ABCC98])47 b(D.)26 b(Applegate,)h(R.)e(Bixb)m(y)-8 b(,)27 b(V.)e(Ch)m(v\023)-45 b(atal,)27 b(and)e(W.)h(Co)s(ok.)38 b(On)25 b(the)g(solution)f(of)i(tra)m(v)m(eling)779 1278 y(salesman)d(problems.)37 b(In)23 b Fa(Pr)-5 b(o)g(c)g(e)g(e)g(dings)29 b(of)e(the)h(International)h(Congr)-5 b(ess)28 b(of)f(Mathemati-)779 1391 y(cians,)32 b(V)-7 b(ol.)33 b(III)p Fw(,)e(pages)g(192{202.)j(Do)s (cumen)m(ta)d(Mathematica,)h(Berlin,)d(1998.)330 1578 y([ABCC01])47 b(D.)41 b(Applegate,)j(R.)c(Bixb)m(y)-8 b(,)43 b(V.)e(Ch)m(v\023)-45 b(atal,)43 b(and)d(W.)h(Co)s(ok.)71 b Fq(TSP)40 b Fw(cuts)g(whic)m(h)f(do)i(not)779 1691 y(conform)k(to)h(the)f(template)h(paradigm.)84 b(In)45 b(M.)h(Junger)e(and)h(D.)h(Naddef,)j(editors,)779 1804 y Fa(Computational)35 b(Combinatorial)g(Optimization)p Fw(,)e(pages)e(261{304.)i(Springer,)c(2001.)330 1992 y([ABGL02])49 b(K.)k(Anstreic)m(her,)58 b(N.)53 b(Brixius,)k(J.-P)-8 b(.)54 b(Goux,)59 b(and)53 b(J.)g(Linderoth.)106 b(Solving)52 b(large)779 2105 y(quadratic)41 b(assignmen)m(t)h(problems)f(on)h (computational)g(grids.)74 b Fa(Math)44 b(Pr)-5 b(o)g(gr)g(amming)779 2218 y(B)p Fw(,)30 b(91\(3\):563{588,)36 b(2002.)330 2405 y([A)m(CG)557 2372 y Fo(+)617 2405 y Fw(99])47 b(G.)c(Ausiello,)i (P)-8 b(.)43 b(Crescenzi,)j(G.)d(Gam)m(b)s(osi,)j(V.)e(Kann,)h(A.)f (Marc)m(hetti-Spaccamela,)779 2518 y(and)34 b(M.)i(Protasi.)55 b Fa(Complexity)39 b(and)f(Appr)-5 b(oximation.)40 b(Combinatorial)g (Optimization)779 2631 y(Pr)-5 b(oblems)33 b(and)h(their)f(Appr)-5 b(oximability)35 b(Pr)-5 b(op)g(erties)p Fw(.)43 b(Springer-V)-8 b(erlag,)29 b(Berlin,)g(1999.)330 2819 y([AHS00])122 b(E.)56 b(Arkin,)63 b(R.)57 b(Hassin,)62 b(and)57 b(M.)g(Sviridenk)m (o.)117 b(Appro)m(ximating)56 b(the)h(maxim)m(um)779 2932 y(quadratic)29 b(assignmen)m(t)i(problem.)39 b Fa(Inform.)33 b(Pr)-5 b(o)g(c)g(ess.)34 b(L)-5 b(ett.)p Fw(,)31 b(77\(1\):13{16,)k (2000.)330 3119 y([AHU74])105 b(A.)43 b(V.)h(Aho,)i(J.)d(E.)g(Hop)s (croft,)k(and)c(J.)g(D.)g(Ullman.)77 b Fa(The)45 b(design)f(and)i (analysis)g(of)779 3232 y(c)-5 b(omputer)34 b(algorithms)p Fw(.)43 b(Addison-W)-8 b(esley)29 b(Pub.)h(Co.,)h(Reading,)f(Mass.,)h (1974.)330 3420 y([Aro98])160 b(S.)28 b(Arora.)40 b(P)m(olynomial-time) 28 b(appro)m(ximation)g(sc)m(hemes)h(for)g(euclidean)e Fq(TSP)h Fw(and)h(other)779 3533 y(geometric)i(problems.)39 b Fa(Journal)33 b(of)g(the)g(A)n(CM)p Fw(,)d(45\(5\):753{782,)36 b(1998.)330 3720 y([AZ01])185 b(E.)30 b(Angel)g(and)g(V.)h(Zissimop)s (oulos.)37 b(On)30 b(the)g(landscap)s(e)f(ruggedness)h(of)h(the)f (quadratic)779 3833 y(assignmen)m(t)35 b(problem.)56 b Fa(The)-5 b(or)g(et.)39 b(Comput.)g(Sci.)p Fw(,)e (263\(1-2\):159{172,)44 b(2001.)59 b(Com)m(bi-)779 3946 y(natorics)30 b(and)g(computer)g(science)g(\(P)m(alaiseau,)h(1997\).) 330 4134 y([Bar])254 b(A.)28 b(Barvinok.)39 b(Estimating)28 b Fr(l)1821 4101 y Ff(1)1923 4134 y Fw(norms)g(b)m(y)g Fr(l)2348 4101 y Fo(2)p Fp(k)2454 4134 y Fw(norms)f(for)h(functions)f (on)h(orbits.)39 b Fa(F)-7 b(oun-)779 4247 y(dations)34 b(of)f(Computational)i(Mathematics)p Fw(,)d(\(to)g(app)s(ear\).)330 4434 y([BC)-53 b(\030)13 b(K94])108 b(R.)36 b(Burk)-5 b(ard,)38 b(E.)f(C)-53 b(\030)13 b(ela,)38 b(and)e(B.)i(Klinz.)58 b(On)36 b(the)h(biquadratic)e(assignmen)m(t)h(problem.)779 4547 y(In)h Fa(Quadr)-5 b(atic)41 b(Assignment)f(and)h(R)-5 b(elate)g(d)42 b(Pr)-5 b(oblems)40 b(\(New)g(Brunswick,)i(NJ,)d(1993\)) p Fw(,)779 4660 y(pages)31 b(117{146.)i(Amer.)e(Math.)g(So)s(c.,)g(Pro) m(vidence,)f(RI,)g(1994.)330 4848 y([BC)-53 b(\030)13 b(PP99])55 b(R.)32 b(Burk)-5 b(ard,)33 b(E.)g(C)-53 b(\030)13 b(ela,)33 b(P)-8 b(.)34 b(P)m(ardalos,)f(and)f(L.)h(Pitsoulis.)46 b(The)32 b(quadratic)g(assignmen)m(t)779 4961 y(problem.)46 b(In)32 b(D.-Z.)i(Du)e(and)h(P)-8 b(.M.)34 b(P)m(ardalos,)f(editors,)g Fa(Handb)-5 b(o)g(ok)37 b(of)e(Combinatorial)779 5074 y(Optimization)f(vol.)f(3)p Fw(,)e(pages)g(241{337.)i(Klu)m(w)m(er)d (Academic)h(Publishers,)c(1999.)330 5261 y([BCSS97])77 b(L.)22 b(Blum,)i(F.)f(Cuc)m(k)m(er,)i(M.)f(Sh)m(ub,)f(and)f(S.)h (Smale.)37 b Fa(Complexity)27 b(and)g(R)-5 b(e)g(al)27 b(Computation)p Fw(.)779 5374 y(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(1997.)p eop %%Page: 112 120 112 119 bop 2047 -210 a Fw(112)330 91 y([BKR97])107 b(R.)22 b(Burk)-5 b(ard,)23 b(S.)f(Karisc)m(h,)i(and)e(F.)h(Rendl.)36 b Fq(QAPLIB)21 b Fw(-)h(a)h(quadratic)f(assignmen)m(t)g(problem)779 204 y(library)-8 b(.)39 b Fa(Journal)34 b(of)f(Glob)-5 b(al)34 b(Optimization)p Fw(,)f(10:391{403,)i(1997.)43 b(Problems)30 b(and)g(solu-)779 317 y(tions)f(main)m(tained)h(on-line)e (at:)42 b Fq(http://www.opt.math.tu-g)o(raz.)o(ac.)o(at/q)o(apli)o(b)p Fw(.)330 505 y([Bl\177)-45 b(a02])174 b(M.)42 b(Bl\177)-45 b(aser.)73 b(An)41 b(8/13-appro)m(ximation)i(algorithm)d(for)h(the)h (asymmetric)f(maxim)m(um)779 618 y Fq(TSP)p Fw(.)f(In)30 b Fa(Pr)-5 b(o)g(c)g(e)g(e)g(dings)35 b(of)e(the)g(Thirte)-5 b(enth)35 b(A)n(nnual)d(A)n(CM-SIAM)f(Symp)-5 b(osium)34 b(on)g(Dis-)779 731 y(cr)-5 b(ete)32 b(A)n(lgorithms)p Fw(,)g(pages)f(64{73.)i(Asso)s(ciation)d(for)g(Computing)f(Mac)m (hinery)-8 b(,)31 b(2002.)330 918 y([BMCP98])47 b(A.)31 b(Br)s(\177)-48 b(ungger,)31 b(A.)g(Marzetta,)i(J.)d(Clausen,)g(and)g (M.)h(P)m(erregaard.)42 b(Solving)29 b(large)i(scale)779 1031 y(quadratic)23 b(assignmen)m(t)g(problems)f(in)h(parallel)f(with)g (the)i(searc)m(h)h(library)c(ZRAM.)38 b Fa(Jour-)779 1144 y(nal)33 b(of)g(Par)-5 b(al)5 b(lel)34 b(and)f(Distribute)-5 b(d)33 b(Computing)p Fw(,)f(50:157{66,)i(1998.)330 1332 y([CJMZ01])57 b(J.)34 b(Cirasella,)g(D.)h(S.)f(Johnson,)h(L.)f(A.)h (McGeo)s(c)m(h,)j(and)c(W.)h(Zhang.)52 b(The)34 b(asymmetric)779 1445 y(tra)m(v)m(eling)43 b(salesman)f(problem:)64 b(Algorithms,)45 b(instance)e(generators,)k(and)c(tests.)79 b(In)779 1558 y(A.)32 b(L.)f(Buc)m(hsbaum)g(and)g(J.)h(Sno)s(eyink,)e(editors,)i Fa(ALENEX)f(2001)36 b(Pr)-5 b(o)g(c)g(e)g(e)g(dings)p Fw(,)34 b(pages)779 1670 y(32{59.)e(Springer)d(Lecture)h(Notes)i(in)d (Computer)g(Science)i(2153,)h(2001.)330 1858 y([Dia88])169 b(P)-8 b(.)35 b(Diaconis.)55 b Fa(Gr)-5 b(oup)39 b(R)-5 b(epr)g(esentations)40 b(in)d(Pr)-5 b(ob)g(ability)38 b(and)g(Statistics)p Fw(.)56 b(Institute)35 b(of)779 1971 y(Mathematical)c(Statistics,)f(Ha)m(yw)m(ard,)h(CA,)g(1988.)330 2159 y([FH91])182 b(W.)37 b(F)-8 b(ulton)36 b(and)g(J.)h(Harris.)58 b Fa(R)-5 b(epr)g(esentation)42 b(The)-5 b(ory:)55 b(A)38 b(First)h(Course)p Fw(.)60 b(Springer-)779 2271 y(V)-8 b(erlag,)31 b(New)f(Y)-8 b(ork,)32 b(1991.)330 2459 y([GJ83])191 b(I.)38 b(Goulden)g(and)h(D.)g(M.)h(Jac)m(kson.)66 b Fa(Combinatorial)44 b(Enumer)-5 b(ation)p Fw(.)67 b(John)38 b(Wiley)g(&)779 2572 y(Sons,)29 b(Inc.,)i(New)g(Y)-8 b(ork,)31 b(1983.)330 2760 y([GL98])181 b(F.)34 b(Glo)m(v)m(er)h(and)f (M.)h(Laguna.)52 b(T)-8 b(abu)34 b(searc)m(h.)53 b(In)33 b Fa(Handb)-5 b(o)g(ok)38 b(of)e(Combinatorial)j(Opti-)779 2873 y(mization,)33 b(V)-7 b(ol.)33 b(3)p Fw(,)e(pages)g(621{757.)j (Klu)m(w)m(er)29 b(Acad.)i(Publ.,)e(Boston,)j(MA,)f(1998.)330 3060 y([GW70])145 b(G.W.)31 b(Gra)m(v)m(es)h(and)d(A.)h(B.)h(Whinston.) 39 b(An)30 b(algorithm)f(for)h(the)h(quadratic)e(assignmen)m(t)779 3173 y(problem.)39 b Fa(Management)33 b(Scienc)-5 b(e)p Fw(,)30 b(17\(7\):452{471,)36 b(1970.)330 3361 y([GY02])170 b(G.)36 b(Gutin)f(and)g(A.)h(Y)-8 b(eo.)58 b(P)m(olynomial)35 b(appro)m(ximation)f(algorithms)h(for)h(the)g Fq(TSP)e Fw(and)779 3474 y(the)39 b Fq(QAP)f Fw(with)g(a)h(factorial)g (domination)e(n)m(um)m(b)s(er.)65 b Fa(Discr)-5 b(ete)41 b(Applie)-5 b(d)41 b(Mathematics)p Fw(,)779 3587 y(119\(1-2\):107{116,) c(2002.)330 3774 y([HR00])174 b(R.)28 b(Hassin)f(and)h(S.)g (Rubinstein.)37 b(Better)29 b(appro)m(ximations)e(for)h(max)h Fq(TSP)p Fw(.)39 b Fa(Information)779 3887 y(Pr)-5 b(o)g(c)g(essing)33 b(L)-5 b(etters)p Fw(,)32 b(75\(4\):181{186,)k(2000.)330 4075 y([JK81])191 b(G.)28 b(James)g(and)g(A.)g(Kerb)s(er.)38 b Fa(The)31 b(R)-5 b(epr)g(esentation)33 b(The)-5 b(ory)32 b(of)f(the)g(Symmetric)g(Gr)-5 b(oup)p Fw(.)779 4188 y(Addison-W)d(esley)30 b(Publishing)e(Co.,)33 b(Reading,)e(Mass.,)i (1981.)46 b(With)31 b(a)h(forew)m(ord)g(b)m(y)f(P)-8 b(.)779 4301 y(M.)30 b(Cohn,)g(With)g(an)g(in)m(tro)s(duction)f(b)m(y)h (G.)h(de)f(B.)h(Robinson.)330 4488 y([Kar72])157 b(R.)40 b(M.)h(Karp.)69 b(Reducibilit)m(y)37 b(among)k(com)m(binatorial)e (problems.)69 b(In)39 b Fa(Complexity)44 b(of)779 4601 y(Computer)28 b(Computations)i(\(Pr)-5 b(o)g(c.)28 b(Symp)-5 b(os.,)30 b(IBM)d(Thomas)i(J.)e(Watson)h(R)-5 b(es.)28 b(Center,)779 4714 y(Y)-7 b(orktown)33 b(Heights,)g(N.Y.,)e(1972\))p Fw(,)i(pages)e(85{103,)i(New)e(Y)-8 b(ork,)31 b(1972.)h(Plen)m(um.)330 4902 y([KB57])174 b(T.)30 b(C.)g(Ko)s(opmans)g(and)f(M.)i(J.)g(Bec)m (kmann.)41 b(Assignmen)m(t)30 b(problems)f(and)h(the)g(lo)s(cation)779 5015 y(of)g(economic)h(activit)m(y)-8 b(.)42 b Fa(Ec)-5 b(onometric)g(a)p Fw(,)32 b(pages)f(53{76,)i(1957.)330 5202 y([Kuh55])136 b(H.)31 b(W.)h(Kuhn.)43 b(The)31 b(Hungarian)f (metho)s(d)h(for)g(the)h(assignmen)m(t)f(problem.)42 b Fa(Naval)34 b(R)-5 b(es.)779 5315 y(L)g(o)g(gist.)33 b(Quart.)p Fw(,)e(2:83{97,)j(1955.)p eop %%Page: 113 121 113 120 bop 2047 -210 a Fw(113)330 91 y([La)m(w63])144 b(E.)26 b(L.)g(La)m(wler.)39 b(The)25 b(quadratic)h(assignmen)m(t)g (problem.)37 b Fa(Management)30 b(Scienc)-5 b(e)p Fw(,)26 b(9:586{)779 204 y(599,)31 b(1963.)330 392 y([LP92])190 b(Y.)27 b(Li)e(and)h(P)-8 b(.)28 b(M.)f(P)m(ardalos.)39 b(Generating)27 b(quadratic)f(assignmen)m(t)h(test)g(problems)e(with) 779 505 y(kno)m(wn)37 b(optimal)f(p)s(erm)m(utations.)61 b Fa(Computational)42 b(Optimization)f(and)f(Applic)-5 b(ations)p Fw(,)779 618 y(1:163{184,)34 b(1992.)330 805 y([LPR94])123 b(Y.)33 b(Li,)g(P)-8 b(.)33 b(M.)h(P)m(ardalos,)g(and)e (M.)i(G.)f(C.)g(Resende.)48 b(A)33 b(greedy)h(randomized)e(adaptiv)m(e) 779 918 y(searc)m(h)f(pro)s(cedure)e(for)h(the)h(quadratic)e(assignmen) m(t)i(problem.)39 b(pages)31 b(237{261,)i(1994.)330 1106 y([LS])291 b(M.)41 b(Lew)m(enstein)f(and)g(M.)h(Sviridenk)m(o.)70 b(An)40 b(5/8-appro)m(ximation)h(algorithm)f(for)h(the)779 1219 y(maxim)m(um)29 b(assymetric)h Fq(TSP)p Fw(.)40 b(Submitted.)330 1406 y([MP91])164 b(D.)40 b(L.)g(Miller)e(and)h(J.)h (F.)g(P)m(ekn)m(y)-8 b(.)70 b(Exact)41 b(solution)d(of)i(large)g (asymmetric)f(tra)m(v)m(elling)779 1519 y(salesman)29 b(problems.)39 b Fa(Scienc)-5 b(e)p Fw(,)31 b(231\(4995\):754{761)q(,) 36 b(1991.)330 1707 y([Mur38])139 b(F.)23 b(D.)h(Murnaghan.)37 b Fa(The)26 b(The)-5 b(ory)28 b(of)e(Gr)-5 b(oup)27 b(R)-5 b(epr)g(esentations)p Fw(.)41 b(Johns)22 b(Hopkins)g(Press,)779 1820 y(Baltimore,)30 b(1938.)42 b(Reprin)m(ted)30 b(b)m(y)g(Do)m(v)m (er)i(Publications)c(Inc.,)j(New)f(Y)-8 b(ork,)31 b(1963.)330 2007 y([NVR68])106 b(C.)34 b(E.)g(Nugen)m(t,)j(T.)d(E.)g(V)-8 b(ollmann,)34 b(and)g(J.)g(Ruml.)51 b(An)34 b(exp)s(erimen)m(tal)f (comparison)h(of)779 2120 y(tec)m(hniques)k(for)h(the)g(assignmen)m(t)g (of)g(facilities)e(to)j(lo)s(cations.)66 b Fa(Op)-5 b(er)g(ations)42 b(R)-5 b(ese)g(ar)g(ch)p Fw(,)779 2233 y(16\(1\):150{173,)36 b(1968.)330 2421 y([Que86])147 b(M.)27 b(Queyranne.)39 b(P)m(erformance)28 b(ratio)f(of)g(p)s(olynomial)e(heuristics)g(for)i (triangle)f(inequal-)779 2534 y(it)m(y)k(quadratic)g(assignmen)m(t)g (problems.)39 b Fa(Op)-5 b(er.)33 b(R)-5 b(es.)33 b(L)-5 b(ett.)p Fw(,)31 b(4\(5\):231{234,)36 b(1986.)330 2721 y([Ser84])182 b(A.)41 b(I.)h(Serdyuk)m(o)m(v.)73 b(An)42 b(algorithm)e(with)g(an)i(estimate)g(for)f(the)h(tra)m(v)m(eling)f (salesman)779 2834 y(problem)20 b(of)j(the)g(maxim)m(um.)36 b Fa(Upr)-5 b(avlyaemye)27 b(Sistemy)p Fw(,)e(25:80{86,)j(1984.)39 b(\(in)22 b(Russian\).)330 3022 y([V)-8 b(em98])133 b(R.)27 b(R.)g(V)-8 b(em)m(ugan)m(ti.)41 b(Applications)25 b(of)i(set)h(co)m(v) m(ering,)h(set)f(pac)m(king)f(and)f(set)i(partitioning)779 3135 y(mo)s(dels:)50 b(a)37 b(surv)m(ey)-8 b(.)57 b(In)36 b Fa(Handb)-5 b(o)g(ok)39 b(of)f(Combinatorial)j(Optimization,)f(V)-7 b(ol.)38 b(1)p Fw(,)g(pages)779 3248 y(573{746.)33 b(Klu)m(w)m(er)d (Acad.)h(Publ.,)e(Boston,)i(MA,)g(1998.)330 3435 y([Y)-8 b(e99])209 b(Y.)30 b(Y)-8 b(e.)41 b(Appro)m(ximating)29 b(quadratic)g(programming)g(with)g(b)s(ound)f(and)h(quadratic)g(con-) 779 3548 y(strain)m(ts.)40 b Fa(Math.)33 b(Pr)-5 b(o)g(gr)g(amming)p Fw(,)33 b(84\(2\):219{226,)j(1999.)p eop %%Page: 0 122 0 121 bop 1801 606 a Ft(ABSTRA)m(CT)634 897 y Fv(The)34 b(Distributions)c(of)i(V)-8 b(alues)33 b(in)f(Com)m(binatorial)d (Optimization)g(Problems)2063 1188 y(b)m(y)1780 1389 y(T)-8 b(amon)32 b(Stephen)330 1980 y(Chair:)43 b(Alexandre)33 b(I.)g(Barvinok)471 2563 y(W)-8 b(e)41 b(study)h(the)e(distribution)f (of)h(ob)5 b(jectiv)m(e)41 b(function)f(v)-5 b(alues)40 b(of)g(a)g(com)m(binatorial)d(opti-)330 2780 y(mization)32 b(problem)h(de\014ned)j(on)f(a)f(group,)h(fo)s(cusing)f(on)h(the)g (Quadratic)e(Assignmen)m(t)i(Prob-)330 2996 y(lem)26 b(\()p Fk(QAP)p Fv(\),)j(and)f(its)f(sp)s(ecial)f(case,)k(the)e(T)-8 b(ra)m(v)m(eling)27 b(Salesman)g(Problem)g(\()p Fk(TSP)p Fv(\).)h(F)-8 b(or)27 b(these)i(t)m(w)m(o)330 3213 y(problems,)40 b(w)m(e)g(estimate)e(the)h(fraction)f(of)h(p)s(erm)m(utations)f Fl(\033)43 b Fv(suc)m(h)d(that)f Fl(f)11 b Fv(\()p Fl(\033)t Fv(\))38 b(lies)g(within)g(a)330 3430 y(giv)m(en)h(neigh)m(b)s(orho)s (o)s(d)f(of)g(the)i(optimal)c(v)-5 b(alue)38 b(of)h Fl(f)11 b Fv(,)40 b(and)f(relate)g(the)g(optimal)d(v)-5 b(alue)39 b(to)f(the)330 3646 y(a)m(v)m(erage)e(v)-5 b(alue)34 b(of)h Fl(f)46 b Fv(o)m(v)m(er)36 b(a)f(neigh)m(b)s(orho)s(o)s(d)e(of)i (the)g(optimal)e(p)s(erm)m(utation.)49 b(W)-8 b(e)36 b(describ)s(e)f(a)330 3863 y(natural)j(class)h(of)f Fk(QAP)i Fv(functions)f(\(whic)m(h)g(includes,)i(for)d(example,)i(the)f(ob)5 b(jectiv)m(e)40 b(function)330 4080 y(in)33 b(the)h(asymmetric)f(T)-8 b(ra)m(v)m(eling)33 b(Salesman)g(Problem\))f(with)i(a)f(relativ)m(e)g (abundance)i(of)e(near-)330 4296 y(optimal)22 b(p)s(erm)m(utations.)40 b(Also,)26 b(w)m(e)f(iden)m(tify)f(a)h(large)f(class)g(of)h(functions)f Fl(f)36 b Fv(with)24 b(the)h(prop)s(ert)m(y)330 4513 y(that)36 b(p)s(erm)m(utations)g(close)g(to)g(the)h(optimal)d(p)s(erm)m (utation)h(in)g(the)i(Hamming)d(metric)i(of)f(the)330 4730 y(symmetric)c(group)h Fl(S)1140 4745 y Fp(n)1219 4730 y Fv(tend)g(to)g(pro)s(duce)g(near)g(optimal)d(v)-5 b(alues)32 b(of)f Fl(f)43 b Fv(\(suc)m(h)33 b(is,)f(for)f(example,)330 4946 y(the)45 b(ob)5 b(jectiv)m(e)46 b(function)e(in)h(the)g(symmetric) f(T)-8 b(ra)m(v)m(eling)44 b(Salesman)g(Problem\).)80 b(W)-8 b(e)45 b(sho)m(w)330 5163 y(examples)33 b(of)g Fk(QAP's)h Fv(where)g(just)f(the)h(opp)s(osite)e(happ)s(ens:)46 b(an)33 b(a)m(v)m(erage)h(p)s(erm)m(utation)d(in)h(the)330 5380 y(vicinit)m(y)24 b(of)h(the)h(optimal)d(p)s(erm)m(utation)h(ma)m (y)h(b)s(e)h(m)m(uc)m(h)f(w)m(orse)i(than)f(an)f(a)m(v)m(erage)h(p)s (erm)m(utation)p eop %%Page: 1 123 1 122 bop 2092 -210 a Fw(1)330 91 y Fv(in)32 b(the)h(whole)f(group)h Fl(S)1225 106 y Fp(n)1272 91 y Fv(.)471 308 y(W)-8 b(e)32 b(in)m(terpret)g(our)g(results)g(algorithmically)-8 b(,)27 b(obtaining)j(guaran)m(tees)j(for)e(simple)f(p)s(olyno-)330 525 y(mial)g(and)k(non-p)s(olynomial)29 b(algorithms,)i(and)i(in)g(the) g(con)m(text)i(of)d(heuristics.)46 b(Additionally)-8 b(,)330 741 y(w)m(e)28 b(compare)f(our)g(results)h(to)f(distributional) d(statistics)j(obtained)f(via)h(computational)d(exp)s(er-)330 958 y(imen)m(ts.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF