Talk:Time hierarchy theorem
 | Time hierarchy theorem received a peer review by Wikipedia editors, which is now archived. It may contain ideas you can use to improve this article. |
Room for Creativity
As a construcive suggestion, I am trying to encourage some creativity in this article and a few others on Wikipedia having to do with computer science. I run a search engine with a goal of curing cancer for children.
Here is my suggestion.A survey of the Latin and Greek meanings of the 'Polynomial' on the Internet Search Engine MeAmI (http://www.meami.org/) in UTF-16 returns ten results to one page in 0.30 seconds. The results are below and to the right with comments in the left margin:
|Results 1 - 10 for polynomial latin or greek meaning. (0.30 seconds) note: [prefix] -http://www = |
|History of the Word "Polynomial" - Math Forum - Ask Dr. Math2 posts - Last post: Oct 20, 2006
suffix: '-nomy, 'nomos' means 'law' |What I find is -nomy, as in astronomy; that comes from Greek nomos, meaning law. But I'm not at all confident that polynomial comes from ...
|mathforum.org/library/drmath/view/69475.html
{not displayed} |
|Greek section and bibliography of A Selection of Latin and Greek ...
|Greek section of an online book containing over 400 Latin Roots and Greek Roots, definitions, and examples. ...
'many' |polys, many, polynomial, polymer ...
'Naus' or 'Navis', boats | In Greek NAUS remained NAUS, while in Latin NAUS became NAVIS -- both meaning BOAT. ...
'root' +www = |www.abasiccurriculum.com/homeschool/roots/athens/
|
|Latin Roots Lesson Plans - LessonCorner Analyze word parts to determine the meaning of unfamiliar words. ...
| The students create the Family names using provided Latin and Greek root words as ...
| The polynomial system, that was then in use, consisted of several Latin, Greek or ...
'Latin_Roots' |www.lessoncorner.com/English/Linguistics/Latin_Roots
|
'seven' |heptagon: Definition from Answers.com[Latin heptagōnon , from Greek heptagōnos , having seven. ...
2, 4, 16, 256, 65,536, 4,294,967,296 | whereas the degree of the minimal polynomial for a constructible number must be a power of 2. ...
18,446,744,073,709,551,616 |www.answers.com/topic/heptagon
(powers of two) |
'sample words' |Ninth Grade Word "Cells" (Greek and Latin Root Words) Greek and Latin Root Words. List #1. Cell. Meaning. Sample Words. act. do. action, actor, react, transact ....
6 appear before | polynomial, polygamy, polygon ...
|www.davis.k12.ut.us/ffjh/Thompson/cells.htm
(6 before 7 or more) |
|mathematics - definition of mathematics by the Free Online ...
'[C14'... '(n)' ... '(adj)' | [C14 mathematik (n), via Latin from Greek (adj), from mathēma a science, ...
(fourth dot) |. multinomial, polynomial - a mathematical function that is the sum of a number ...
|www.thefreedictionary.com/mathematics
|
From 'arithmetic terms' (-2+) |Origins of some arithmetic terms-2Plus
(2 + 4) 'more' |(as in two plus four) comes from the early Latin word meaning "more". ...
'many names' | The word is a hybrid of Greek and Latin roots. Polynomial means "many names" ...
|www.pballew.net/arithme3.html
|
|Algorithm - Wikipedia, the free encyclopedia We can derive clues to the issues involved and an informal meaning of the ...
|.. For example, the algorithms that run in polynomial time suffice for many ...
| via European Latin translation of al-Khwarizmi's name into algorithm by the 18th century. ...
| The work of the ancient Greek geometers, Persian mathematician ...
|en.wikipedia.org/wiki/Algorithm
|
|Z English Greek Form Latin French Letter Zeta Voiced **z Symbol In Vulgar Latin the Greek Z seems to have been pronounced as dy and later y ...
| for six European langua for the meaning of all the above phonetic symbols. ...
| Characteristic function · Characteristic polynomial · Characteristic equation ...
|www.economicexpert.com/a/Character:Z.htm
|
|MULTIPLY COMPOSABLE HOMOGENEOUS POLYNOMIALS.*by ET Bell - 1930
|definition of composition as applied to homogeneous polynomials.t ...
|. instance, the meaning will, however, always be unambiguous. As in tensor analysis, a repeated Latin letter in a product occurring ...
| Capital Greek letters, as Of, denote elements in algebraic number rings ; ...
|www.jstor.org/stable/1968150
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Seven boats or roots of polynomial constructable numbers consist of seven powers of two. The first seven powers of two are the numbers:
ROOT 1 2.0000000000000000000000000000000000000000000000000 (49 ZEROES)
ROOT 2 4.0000000000000000000000000000000000000000000000000 (49 ZEROES)
ROOT 3 1.600000000000000000000000000000000000000000000000X (48 ZEROES)
ROOT 4 2.560000000000000000000000000000000000000000000000X (47 ZEROES)
R00T 5 6.553600000000000000000000000000000000000000000000X (45 ZEROES)
ROOT 6 4.429496729600000000000000000000000000000000000000X (39 ZEROES)
ROOT 7 1.844674407370955161600000000000000000000000000000X (30 ZEROES)
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RIGHT-ARROW
(C) 2010. MeAmI.org. M. Michael Musatov. All Rights Reserved. Posted with permission of the author. Note: the above is Intellectual Property MeAmI.org. (But only to campaign the Clay Mathematics Foundation and donate a million dollars to http://www.alexslemonade.org/ in support of a cure to childhood cancer.) Page: 1 1 1 0 bop 1208 458 a Fn(Con)-6 b(ten)g(ts)71 2077 y Fm(1)180 b(In)m(tro)s(duction)2717 b Fl(p)-5 b(age)38 b Fk(2)303 2309 y Fm(Bibliograph)m(y)2917 b Fk(7)1938 6011 y(1)p eop %%Page: 2 2 2 1 bop 1464 159 a Fj(1)p -118 226 3260 3 v 834 458 a Fn(In)-6 b(tro)6 b(duction)-118 1961 y Fk(Linear)22 b(Programming)g (deals)h(with)f(the)h(optimization)g(of)g(linear)f(functions)f(sub)5 b(ject)23 b(to)h(linear)e(constrain)m(ts.)-118 2077 y(F)-8 b(or)31 b(example,)1240 2341 y(minimize)119 b(3)p Fi(x)20 b Fk(+)g(4)p Fi(y)1240 2457 y Fk(sub)5 b(ject)30 b(to)84 b(2)p Fi(x)20 b Fk(+)g(3)p Fi(y)86 b Fh(\024)d Fk(7)1720 2573 y(3)p Fi(x)20 b Fh(\000)g Fk(4)p Fi(y)86 b Fh(\024)d Fk(4)-27 2738 y(Man)m(y)33 b(problems)f(can)h(b)s(e)f(form)m(ulated)g (naturally)f(as)j(linear)d(programs.)48 b(W)-8 b(e)34 b(will)c(see)k(some)f(examples)-118 2854 y(b)s(elo)m(w.)40 b(Linear)29 b(programming)g(giv)m(es)i(a)g(general)f(metho)s(d)g(for)g (solving)f(problems)g(in)g(p)s(olynomial)f(time.)-118 3070 y Fm(Theorem)33 b(1)h Fl(Line)-5 b(ar)33 b(pr)-5 b(o)g(gr)g(ams)35 b(c)-5 b(an)32 b(b)-5 b(e)32 b(solve)-5 b(d)33 b(in)f(time)g(p)-5 b(olynomial)35 b(in)d(the)g(numb)-5 b(er)32 b Fi(m)g Fl(of)g(c)-5 b(onstr)g(aints,)-118 3186 y(the)30 b(numb)-5 b(er)30 b Fi(n)g Fl(of)g(variables,)h(and)f(the)h (total)g(size)e Fi(L)h Fl(of)g(the)g(c)-5 b(o)g(e\016cients.)42 b(The)30 b(size)g(of)g(a)g(c)-5 b(o)g(e\016cient)30 b(is)g(the)-118 3302 y(lo)-5 b(garithm)35 b(of)e Fk(1)g Fl(plus)g(its)g(absolute)g (value.)-27 3473 y Fk(In)27 b(In)m(teger)i(Linear)e(Programming)g(the)h (v)-5 b(ariables)27 b(are)i(restricted)e(to)i(tak)m(e)h(in)m(tegral)d (v)-5 b(alues.)40 b(F)-8 b(or)29 b(exam-)-118 3589 y(ple,)1232 3853 y(minimize)119 b(3)p Fi(x)21 b Fk(+)f(4)p Fi(y)1232 3969 y Fk(sub)5 b(ject)30 b(to)84 b(2)p Fi(x)21 b Fk(+)f(3)p Fi(y)86 b Fh(\024)d Fk(7)1712 4085 y(3)p Fi(x)21 b Fh(\000)f Fk(4)p Fi(y)86 b Fh(\024)d Fk(4)1874 4201 y Fi(x;)15 b(y)91 b Fh(2)d Fg(Z)-118 4407 y Fm(Theorem)26 b(2)h Fl(The)g(de)-5 b(cision)27 b(pr)-5 b(oblem)28 b(for)f(inte)-5 b(ger)26 b(line)-5 b(ar)28 b(pr)-5 b(o)g(gr)g(amming)29 b(is)e(NP-c)-5 b(omplete.)40 b(The)26 b(input)h(for)-118 4524 y(the)34 b(de)-5 b(cision)36 b(pr)-5 b(oblem)35 b(is)g(an)f(inte)-5 b(ger)35 b(line)-5 b(ar)35 b(pr)-5 b(o)g(gr)g(am)37 b(and)e(an)g(inte)-5 b(ger)34 b Fi(k)k Fl(and)d(the)f(question)h(is)f(whether)-118 4640 y(the)f(pr)-5 b(o)g(gr)g(am)36 b(has)d(a)g(fe)-5 b(asible)33 b(solution)h(with)g (obje)-5 b(ctive)32 b(value)h(at)g(most)h Fi(k)s Fl(.)-27 4810 y Fk(The)22 b(NP-completeness)h(of)g(the)g(decision)e(problem)h (for)g(in)m(teger)h(linear)e(programming)h(mak)m(es)i(it)e(unlik)m(ely) -118 4926 y(that)34 b(there)h(is)e(a)h(p)s(olynomial)d(time)j (algorithm)f(for)h(in)m(teger)g(linear)e(programming.)50 b(Nev)m(ertheless,)36 b(there)-118 5043 y(are)e(general)f(metho)s(ds)g (for)h(solving)e(ILP)h(exactly)h(and)f(appro)m(ximately)g(and)g(these)h (metho)s(ds)f(are)h(among)-118 5159 y(the)d(most)h(p)s(o)m(w)m(erful)d (metho)s(ds)h(for)h(attac)m(king)i(NP-complete)f(problems.)41 b(The)30 b(NP-completeness)i(of)f(ILP)-118 5275 y(implies)i(that)j(ev)m (ery)g(problem)e(in)h(NP)g(can)h(b)s(e)f(reduced)g(to)h(an)g(ILP)-8 b(.)36 b(More)g(imp)s(ortan)m(t)f(is)f(the)i(fact)h(that)-118 5391 y(man)m(y)30 b(di\016cult)f(optimization)g(problems)g(are)h(form)m (ulated)g(v)m(ery)h(naturally)e(as)h(ILPs.)-27 5507 y(W)-8 b(e)32 b(summarize:)41 b(Linear)30 b(Programming)g(and)g(In)m(teger)i (Linear)e(Programming)g(are)h(extremely)g(p)s(o)m(w)m(er-)1749 6011 y(2)p eop %%Page: 3 3 3 2 bop 71 -214 a Ff(In)n(tro)r(duction)3320 b(3)71 101 y Fk(ful)34 b(algorithmic)f(paradigms:)50 b(man)m(y)35 b(problems)e(are)j(easily)e(form)m(ulated)h(as)h(LPs)e(or)i(ILPs)e(and) h(e\013ectiv)m(e)71 217 y(solution)29 b(algorithms)g(are)i(a)m(v)-5 b(ailable.)162 333 y(W)d(e)31 b(next)g(giv)m(e)g(some)g(examples)e(of)i (linear)e(and)h(in)m(teger)g(linear)f(programs.)71 554 y Fm(Maxim)m(um)41 b(Net)m(w)m(ork)g(Flo)m(w:)k Fk(W)-8 b(e)38 b(ha)m(v)m(e)g(a)f(directed)f(graph)g Fi(G)g Fk(=)f(\()p Fi(V)5 b(;)15 b(E)5 b Fk(\),)40 b(a)d(source)g(no)s(de)f Fi(s)f Fh(2)g Fi(V)21 b Fk(,)38 b(a)71 670 y(sink)d(no)s(de)g Fi(t)h Fh(2)e Fi(V)21 b Fk(,)38 b(and)d(a)i(non-negativ)m(e)g(capacit)m (y)h Fl(c)-5 b(ap)2099 692 y Fe(e)2172 670 y Fk(for)36 b(ev)m(ery)h(edge)g(of)g Fi(e)p Fk(.)59 b(The)36 b(goal)h(is)e(to)i (\014nd)e(a)71 786 y(maxim)m(um)d(\015o)m(w)i(from)f Fi(s)h Fk(to)g Fi(t)p Fk(.)50 b(F)-8 b(or)35 b(a)f(no)s(de)f Fi(v)s Fk(,)i(w)m(e)f(use)f Fl(In)8 b Fk(\()p Fi(v)s Fk(\))34 b(and)f Fl(Out)9 b Fk(\()p Fi(v)s Fk(\))34 b(to)h(denote)f (the)g(set)g(of)g(edges)71 902 y(ending)f(in)h(and)g(emanating)h(from)g Fi(v)s Fk(,)h(resp)s(ectiv)m(ely)-8 b(,)36 b(and)f(for)f(an)h(edge)h Fi(e)f Fk(w)m(e)g(use)g Fi(f)3046 916 y Fe(e)3117 902 y Fk(to)h(denote)f(the)g(\015o)m(w)71 1019 y(along)30 b(the)h(edge.)523 1182 y(maximize)1007 1114 y Fd(P)1103 1209 y Fe(e)p Fc(2)p Fb(O)6 b Fe(ut)p Fa(\()p Fe(s)p Fa(\))1416 1182 y Fi(f)1461 1196 y Fe(e)523 1307 y Fk(sub)f(ject)30 b(to)1003 1239 y Fd(P)1099 1334 y Fe(e)p Fc(2)p Fb(O)6 b Fe(ut)p Fa(\()p Fe(v)r Fa(\))1416 1307 y Fi(f)1461 1321 y Fe(e)1581 1307 y Fk(=)1735 1239 y Fd(P)1831 1334 y Fe(e)p Fc(2)p Fb(I)k Fe(n)p Fa(\()p Fe(v)r Fa(\))2102 1307 y Fi(f)2147 1321 y Fe(e)2266 1307 y Fk(for)30 b(ev)m(ery)i(no)s (de)d Fi(v)g Fh(2)c Fi(V)40 b Fh(n)21 b(f)5 b Fi(s;)15 b(t)5 b Fh(g)1416 1432 y Fi(f)1461 1446 y Fe(e)1581 1432 y Fh(\024)83 b Fl(c)-5 b(ap)1870 1454 y Fe(e)2266 1432 y Fk(for)30 b(ev)m(ery)i(edge)f Fi(e)25 b Fh(2)g Fi(E)1416 1548 y(f)1461 1562 y Fe(e)1581 1548 y Fh(\025)83 b Fk(0)71 1711 y(The)33 b(\014rst)f(family)g(of)h(constrain)m(ts)g(states)i(that) e(w)m(e)h(ha)m(v)m(e)g(\015o)m(w)g(conserv)-5 b(ation)33 b(at)h(all)e(no)s(des)g(distinct)g(from)71 1827 y Fi(s)k Fk(and)f Fi(t)h Fk(and)g(the)g(other)g(constrain)m(ts)h(state)g(that)g (\015o)m(ws)f(are)g(non-negativ)m(e)i(and)d(b)s(ounded)f(b)m(y)i(the)h (edge)71 1943 y(capacities.)k(Sub)5 b(ject)30 b(to)h(these)g(constrain) m(ts)f(w)m(e)h(are)g(maximizing)d(the)j(\015o)m(w)f(out)h(of)f Fi(s)p Fk(.)71 2163 y Fm(Minim)m(um)i Fi(s)p Fm(-)p Fi(t)p Fm(-Cut:)44 b Fk(In)28 b(the)h(setting)g(of)g(the)g(maxim)m(um)f(\015o) m(w)h(problem)f(w)m(e)h(ma)m(y)g(also)g(b)s(e)g(in)m(terested)g(in)71 2280 y(a)e(minim)m(um)d(capacit)m(y)k(cut)f Fi(C)33 b Fk(separating)27 b Fi(s)f Fk(and)g Fi(t)p Fk(.)39 b(A)27 b(set)h Fi(C)33 b Fk(of)27 b(edges)g(is)f(called)g(an)g Fi(s)p Fk(-)p Fi(t)p Fk(-cut)h(if)f(ev)m(ery)i(path)71 2396 y(from)33 b Fi(s)h Fk(to)g Fi(t)g Fk(con)m(tains)g(at)h(least)f (one)g(edge)h(in)d Fi(C)7 b Fk(.)51 b(The)33 b(capacit)m(y)i(of)g Fi(C)40 b Fk(is)33 b(the)h(sum)f(of)h(the)g(capacities)g(of)71 2512 y(the)27 b(edges)g(in)e Fi(C)7 b Fk(.)39 b(The)26 b(minim)m(um)e Fi(s)p Fk(-)p Fi(t)p Fk(-cut)j(problem)e(is)g(easily)h (form)m(ulated)g(as)h(a)g(ILP)-8 b(.)26 b(W)-8 b(e)28 b(ha)m(v)m(e)g(a)f(zero-one)71 2628 y(v)-5 b(ariable)29 b(for)h(ev)m(ery)h(edge)g Fi(e)g Fk(with)e(the)i(in)m(tended)e(seman)m (tics)i(that)f Fi(e)c Fh(2)f Fi(C)37 b Fk(i\013)29 b Fi(y)2824 2642 y Fe(e)2886 2628 y Fk(=)c(1.)787 2792 y(minimize)1267 2724 y Fd(P)1363 2819 y Fe(e)p Fc(2)p Fe(E)1517 2792 y Fl(c)-5 b(ap)1653 2814 y Fe(e)1690 2792 y Fi(y)1735 2806 y Fe(e)787 2908 y Fk(sub)5 b(ject)30 b(to)1459 2840 y Fd(P)1555 2935 y Fe(e)p Fc(2)p Fe(p)1690 2908 y Fi(y)1735 2922 y Fe(e)1854 2908 y Fh(\025)83 b Fk(1)269 b(for)31 b(ev)m(ery)g Fi(s)p Fk(-)p Fi(t)p Fk(-path)f Fi(p)1690 3029 y(y)1735 3043 y Fe(e)1859 3029 y Fh(2)88 b(f)5 b Fk(0)p Fi(;)15 b Fk(1)5 b Fh(g)84 b Fk(for)31 b(ev)m(ery)g(edge)g Fi(e)25 b Fh(2)g Fi(E)71 3192 y Fk(The)h(n)m(um)m (b)s(er)f(of)h(constrain)m(ts)h(is)e(the)i(ILP)e(ab)s(o)m(v)m(e)j(is)d (equal)h(to)h(the)g(n)m(um)m(b)s(er)e(of)h Fi(s)p Fk(-)p Fi(t)p Fk(-paths.)39 b(In)26 b(a)h(graph)f(with)71 3308 y(cycles,)37 b(this)d(n)m(um)m(b)s(er)f(is)i(in\014nite.)52 b(W)-8 b(e)37 b(ma)m(y)e(restrict)g(atten)m(tion)h(to)g(simple)d Fi(s)p Fk(-)p Fi(t)p Fk(-paths.)55 b(The)34 b(n)m(um)m(b)s(er)g(of)71 3424 y(simple)28 b(paths)i(is)g(\014nite,)f(but)h(is)f(in)g(general)i (exp)s(onen)m(tial)e(in)g(the)i(size)f(of)g(the)h(graph.)71 3563 y Fm(Miracle:)69 b Fk(One)44 b(can)h(solv)m(e)g(LPs)f(and)g(ILPs)g (with)f(an)h(exp)s(onen)m(tial)g(or)g(ev)m(en)h(an)g(in\014nite)d(n)m (um)m(b)s(er)h(of)71 3680 y(constrain)m(ts.)162 3842 y(There)c(is)g(an)h(alternativ)m(e)h(form)m(ulation)e(of)h(the)g(minim) m(um)d(cut)k(problem)d(whic)m(h)h(uses)g(only)g(a)i(linear)71 3958 y(n)m(um)m(b)s(er)34 b(of)i(constrain)m(ts.)56 b(W)-8 b(e)37 b(no)m(w)e(ha)m(v)m(e)i(a)f(zero-one)h(v)-5 b(ariable)34 b(for)h(eac)m(h)i(no)s(de)e(and)g(eac)m(h)i(edge;)i(sa)m(y)d Fi(y)3810 3972 y Fe(v)71 4074 y Fk(is)j(the)h(v)-5 b(ariable)39 b(for)h(no)s(de)f Fi(v)k Fk(and)d Fi(y)1386 4088 y Fe(e)1462 4074 y Fk(is)f(the)i(v)-5 b(ariable)38 b(for)i(edge)h Fi(e)p Fk(.)70 b(The)39 b(in)m(tended)g(seman)m(tics)i(is)e(that)71 4191 y Fi(C)34 b Fk(=)27 b Fh(f)5 b Fi(e)31 b Fk(;)f Fi(y)491 4205 y Fe(e)553 4191 y Fk(=)25 b(1)5 b Fh(g)32 b Fk(is)f(an)h Fi(s)p Fk(-)p Fi(t)p Fk(-cut)g(and)f(that)h Fi(y)1712 4205 y Fe(v)1780 4191 y Fk(=)c(1)k(i\013)f Fi(v)k Fk(can)d(b)s(e)f(reac)m(hed)i(from)e Fi(s)g Fk(b)m(y)h(a)g(path) g(not)g(using)71 4307 y(an)e(edge)h(in)e Fi(C)7 b Fk(.)621 4470 y(minimize)1101 4402 y Fd(P)1197 4497 y Fe(e)p Fc(2)p Fe(E)1351 4470 y Fl(c)-5 b(ap)1487 4492 y Fe(e)1524 4470 y Fi(y)1569 4484 y Fe(e)621 4587 y Fk(sub)5 b(ject)30 b(to)110 b Fi(y)1172 4601 y Fe(u)1237 4587 y Fk(+)19 b Fi(y)1372 4601 y Fe(e)1429 4587 y Fh(\000)h Fi(y)1565 4601 y Fe(v)1689 4587 y Fh(\025)82 b Fk(0)270 b(for)30 b(ev)m(ery)h(edge)g Fi(e)26 b Fk(=)e(\()p Fi(u;)15 b(v)s Fk(\))27 b Fh(2)e Fi(E)1524 4703 y(y)1569 4717 y Fe(e)1694 4703 y Fh(2)87 b(f)5 b Fk(0)p Fi(;)15 b Fk(1)5 b Fh(g)85 b Fk(for)30 b(ev)m(ery)h(edge)g Fi(e)26 b Fh(2)e Fi(E)1520 4819 y(y)1565 4833 y Fe(v)1694 4819 y Fh(2)87 b(f)5 b Fk(0)p Fi(;)15 b Fk(1)5 b Fh(g)85 b Fk(for)30 b(ev)m(ery)h(no)s(de)f Fi(v)e Fh(2)d Fi(V)1524 4935 y(y)1569 4949 y Fe(s)1689 4935 y Fk(=)82 b(0)1531 5052 y Fi(y)1576 5066 y Fe(t)1689 5052 y Fk(=)g(1)71 5214 y(W)-8 b(e)31 b(need)g(to)g(argue)f(that)h(the) g(ILP)f(captures)g(the)h Fi(s)p Fk(-)p Fi(t)p Fk(-cut)f(problem.)162 5330 y(Wir)c(\014rst)g(sho)m(w)g(that)i(an)m(y)f(feasible)e(solution)g (de\014nes)h(a)h(cut.)40 b(Consider)25 b(an)m(y)i Fi(s)p Fk(-)p Fi(t)p Fk(-path)g Fi(p)p Fk(.)39 b(Summing)24 b(the)71 5447 y(\014rst)29 b(constrain)m(t)h(for)g(all)f(edges)h Fi(e)c Fh(2)f Fi(P)42 b Fk(giv)m(es)31 b Fi(y)1714 5461 y Fe(s)1769 5447 y Fk(+)1859 5378 y Fd(P)1955 5473 y Fe(e)p Fc(2)p Fe(p)2090 5447 y Fi(y)2135 5461 y Fe(e)2191 5447 y Fh(\000)19 b Fi(y)2326 5461 y Fe(t)2380 5447 y Fh(\025)25 b Fk(0)30 b(and)g(hence)g(with)f(the)h(use)f(of)h Fi(y)3647 5461 y Fe(s)3709 5447 y Fk(=)25 b(0)71 5563 y(and)30 b Fi(y)293 5577 y Fe(t)347 5563 y Fk(=)25 b(1,)544 5495 y Fd(P)640 5590 y Fe(e)p Fc(2)p Fe(p)775 5563 y Fi(y)820 5577 y Fe(e)881 5563 y Fh(\025)g Fk(1.)41 b(The)30 b(set)h(of)g(edges)g(with)e Fi(y)2016 5577 y Fe(e)2077 5563 y Fk(=)c(1)31 b(is)f(th)m(us)g(guaran)m(teed)h(to)g(form)f(an)g Fi(s)p Fk(-)p Fi(t)p Fk(-cut.)162 5679 y(Con)m(v)m(ersely)-8 b(,)36 b(consider)d(an)m(y)h Fi(s)p Fk(-)p Fi(t)p Fk(-cut)h Fi(C)7 b Fk(.)52 b(Setting)34 b Fi(y)1986 5693 y Fe(e)2054 5679 y Fk(=)d(1)k(i\013)f Fi(e)e Fh(2)f Fi(C)7 b Fk(,)35 b Fi(y)2692 5693 y Fe(v)2764 5679 y Fk(=)d(0)i(for)g(all)f(no)s(des)h (reac)m(hable)p eop %%Page: 4 4 4 3 bop -118 -214 a Ff(4)3321 b(In)n(tro)r(duction)-118 101 y Fk(from)38 b Fi(s)h Fk(b)m(y)f(a)h(path)g(not)g(using)e(an)i (edge)g(of)g Fi(C)7 b Fk(,)41 b(and)d Fi(y)1872 115 y Fe(v)1952 101 y Fk(=)h(1)g(for)f(the)h(remaining)e(no)s(des,)j(w)m(e)g (obtain)e(a)-118 217 y(feasible)24 b(solution)h(of)g(the)h(ILP)-8 b(.)26 b(Observ)m(e)f(that)h Fi(y)1564 231 y Fe(s)1626 217 y Fk(=)f(0)h(and)f Fi(y)2010 231 y Fe(t)2065 217 y Fk(=)f(1)i(since)f(ev)m(ery)i(path)e(from)g Fi(s)g Fk(to)i Fi(t)e Fk(con)m(tains)-118 333 y(at)k(least)f(one)g(edge)g(in)f Fi(C)7 b Fk(.)39 b(Consider)26 b(\014nally)g(an)i(inequalit)m(y)e Fi(y)2047 347 y Fe(u)2107 333 y Fk(+)15 b Fi(y)2238 347 y Fe(e)2299 333 y Fh(\025)25 b Fi(y)2440 347 y Fe(v)2509 333 y Fk(where)i Fi(e)f Fk(=)f(\()p Fi(u;)15 b(v)s Fk(\))29 b(is)e(an)h(edge)g(of)-118 450 y Fi(G)p Fk(.)40 b(If)29 b(either)f Fi(y)409 464 y Fe(u)478 450 y Fk(=)d(1)k(or)g Fi(y)803 464 y Fe(v)869 450 y Fk(=)c(0,)k(the)g(inequalit)m(y)e(is)g (certainly)h(satis\014ed.)39 b(So)29 b(assume)f Fi(y)2969 464 y Fe(u)3039 450 y Fk(=)d(0)k(and)f Fi(y)3429 464 y Fe(v)3495 450 y Fk(=)d(1.)-118 566 y(Then)j(there)i(is)e(a)i(path)f (from)f Fi(s)h Fk(to)h Fi(u)f Fk(using)f(no)h(edge)h(in)e Fi(C)7 b Fk(,)29 b(sa)m(y)h Fi(p)p Fk(,)g(and)e(there)i(is)e(no)h(suc)m (h)g(path)g(from)g Fi(s)g Fk(to)-118 682 y Fi(v)s Fk(.)51 b(Consider)31 b(the)j(path)g Fi(p)22 b Fh(\016)h Fi(e)33 b Fk(from)h Fi(s)f Fk(to)h Fi(v)s Fk(.)51 b(It)34 b(m)m(ust)f(con)m (tain)h(an)g(edge)g(in)e Fi(C)40 b Fk(and)33 b(hence)h Fi(e)d Fh(2)g Fi(C)7 b Fk(.)49 b(Th)m(us)-118 798 y Fi(y)-73 812 y Fe(e)-11 798 y Fk(=)25 b(1)30 b(and)g(the)h(inequalit)m(y)d (holds.)-27 915 y(W)-8 b(e)31 b(conclude)f(that)h(an)f(optimal)g (solution)f(of)h(the)h(ILP)f(is)f(a)i(minim)m(um)c(capacit)m(y)32 b Fi(s)p Fk(-)p Fi(t)p Fk(-cut.)-118 1054 y Fm(Miracle:)46 b Fk(W)-8 b(e)34 b(ma)m(y)g(replace)f(the)g(in)m(tegralit)m(y)g (constrain)m(ts)f Fi(y)2081 1068 y Fe(v)2152 1054 y Fh(2)d(f)5 b Fk(0)p Fi(;)15 b Fk(1)5 b Fh(g)34 b Fk(for)f(all)e Fi(v)i Fh(2)c Fi(V)54 b Fk(and)32 b Fi(y)3274 1068 y Fe(e)3340 1054 y Fh(2)d(f)5 b Fk(0)p Fi(;)15 b Fk(1)5 b Fh(g)-118 1170 y Fk(for)33 b(all)g Fi(e)e Fh(2)f Fi(E)38 b Fk(b)m(y)c(the)g(w)m(eak)m(er)h(constrain)m(ts)e(0)e Fh(\024)f Fi(y)1705 1184 y Fe(v)1776 1170 y Fh(\024)h Fk(1)j(for)f(all)f Fi(v)i Fh(2)c Fi(V)54 b Fk(and)33 b(0)e Fh(\024)f Fi(y)2906 1184 y Fe(e)2973 1170 y Fh(\024)g Fk(1)k(for)g(all)e Fi(e)f Fh(2)f Fi(E)-118 1286 y Fk(without)f(c)m (hanging)i(the)f(optimal)f(ob)5 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2264 y Fi(y)1480 2278 y Fe(t)1592 2264 y Fk(=)g(1)-118 2431 y(is)26 b(guaran)m(teed)i(to)g(ha)m(v)m(e)g(an)g(in)m(tegral)e (optimal)g(solution)g(\(more)i(generally)-8 b(,)27 b(LPs)g(with)f (totally)h(unimo)s(dular)-118 2547 y(constrain)m(t)35 b(matrices)h(are)g(guaran)m(teed)g(to)g(ha)m(v)m(e)h(in)m(tegral)e (solutions\);)i(there)f(ma)m(y)g(also)f(b)s(e)g(non-in)m(tegral)-118 2664 y(optimal)29 b(solution.)-118 2803 y Fm(Miracle:)45 b Fk(The)31 b(v)-5 b(alue)32 b(of)g(the)h(maxim)m(um)e(\015o)m(w)h(is)f (equal)h(to)h(the)f(minim)m(um)d(capacit)m(y)34 b(of)e(an)g Fi(s)p Fk(-)p Fi(t)p Fk(-cut,)h(the)-118 2919 y(so-called)f(Max-Flo)m (w-Min-Cut)h(theorem.)48 b(This)31 b(is)h(called)f(a)i(dualit)m(y)f (theorem.)47 b(It)33 b(is)f(a)h(consequence)g(of)-118 3035 y(the)d(strong)h(dualit)m(y)e(theorem)i(of)f(linear)f (programming.)-118 3256 y Fm(Maxim)m(um)d(W)-9 b(eigh)m(t)28 b(Bipartite)f(Matc)m(hing:)47 b Fk(W)-8 b(e)25 b(ha)m(v)m(e)g(a)f (bipartite)f(graph)h Fi(G)h Fk(=)g(\()p Fi(A)3036 3245 y Fk(_)3017 3256 y Fh([)8 b Fi(B)d(;)15 b(E)5 b Fk(\))25 b(in)e(whic)m(h)-118 3372 y(eac)m(h)32 b(edge)h Fi(e)e Fk(carries)g(a)h(real)f(w)m(eigh)m(t)h Fi(w)1268 3386 y Fe(e)1305 3372 y Fk(.)44 b(A)31 b(matc)m(hing)h Fi(M)41 b Fk(is)30 b(a)i(set)g(of)g(edges)g(no)f(t)m(w)m(o)i(of)e(whic)m(h)f (share)h(an)-118 3488 y(edge)d(p)s(oin)m(t.)39 b(The)26 b(goal)i(is)e(to)i(\014nd)e(a)i(matc)m(hing)f(of)g(maximal)g(w)m(eigh)m (t.)40 b(The)27 b(problem)e(is)h(easily)h(form)m(ulated)-118 3604 y(as)37 b(an)g(ILP)-8 b(.)37 b(W)-8 b(e)38 b(use)f(a)g(zero-one)i (v)-5 b(ariable)35 b Fi(x)1513 3618 y Fe(e)1587 3604 y Fk(for)i(eac)m(h)h(edge)g(with)d(the)j(in)m(tended)d(seman)m(tics)j (that)f(the)-118 3721 y(set)32 b(of)g(edges)h(with)d Fi(x)635 3735 y Fe(e)700 3721 y Fk(=)d(1)33 b(form)e(a)h(matc)m(hing.) 46 b(F)-8 b(or)32 b(a)g(no)s(de)g Fi(v)j Fk(w)m(e)d(use)g Fi(\016)s Fk(\()p Fi(v)s Fk(\))h(to)g(denote)f(the)g(set)h(of)f(edges) -118 3837 y(inciden)m(t)d(to)i Fi(v)s Fk(.)803 3983 y(maximize)1353 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Fh(2)g Fi(E)1696 5400 y(x)1748 5414 y Fe(e)1868 5400 y Fh(\025)83 b Fk(0)g(for)30 b(all)g Fi(e)25 b Fh(2)g Fi(E)-118 5563 y Fk(is)44 b(guaran)m(teed)h(to)h(ha)m (v)m(e)g(an)e(in)m(tegral)h(optimal)f(solution;)50 b(there)45 b(ma)m(y)g(also)g(b)s(e)f(non-in)m(tegral)g(optimal)-118 5679 y(solution.)39 b(Again)30 b(the)h(constrain)m(t)f(matrix)g(is)g (totally)g(unimo)s(dular.)p eop %%Page: 5 5 5 4 bop 71 -214 a Ff(In)n(tro)r(duction)3320 b(5)71 101 y Fm(Minim)m(um)48 b(No)s(de)i(Co)m(v)m(ers)g(in)g(Bipartite)f(Graphs:) d Fk(W)-8 b(e)45 b(are)e(in)f(the)i(same)f(situation)g(as)g(in)f(the)71 217 y(previous)34 b(paragraph.)55 b(The)34 b(goal)i(is)f(to)h(assign)e (a)i(non-negativ)m(e)g(n)m(um)m(b)s(er)e Fi(y)2799 231 y Fe(v)2875 217 y Fk(to)i(ev)m(ery)g(no)s(de)e(of)i Fi(G)g Fk(suc)m(h)71 333 y(that)31 b Fi(w)333 347 y Fe(e)395 333 y Fh(\024)25 b Fi(y)536 347 y Fe(a)597 333 y Fk(+)20 b Fi(y)733 348 y Fe(b)798 333 y Fk(for)30 b(ev)m(ery)h(edge)g Fi(e)25 b Fk(=)g Fi(ab)h Fh(2)f Fi(G)30 b 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y(ev)m(ery)38 b(edge)g Fi(e)g Fk(is)e(assigned)h(a)h(cost)g Fi(c)1393 1401 y Fe(e)1430 1387 y Fk(.)62 b(The)37 b(goal)h(is)f(to)h (\014nd)e(a)h(minim)m(um)e(cost)j(tour)g(passing)e(through)71 1503 y(all)d(no)s(des)h(of)h Fi(G)p Fk(.)54 b(W)-8 b(e)35 b(form)m(ulate)g(the)g(problem)e(as)h(a)h(ILP)-8 b(.)35 b(W)-8 b(e)36 b(ha)m(v)m(e)g(a)e(zero-one)j(v)-5 b(ariable)33 b Fi(x)3426 1517 y Fe(e)3497 1503 y Fk(for)i(ev)m(ery)71 1619 y(edge)30 b Fi(e)p Fk(.)41 b(The)29 b(in)m(tended)f(seman)m(tics)i (is)f(that)h(the)g(set)g(of)g(edges)g(with)e Fi(x)2540 1633 y Fe(e)2602 1619 y Fk(=)d(1)30 b(forms)f(an)h(optimal)e(T)-8 b(ra)m(v)m(eling)71 1735 y(Salesman)30 b(T)-8 b(our.)41 b(F)-8 b(or)31 b(a)g(set)g Fi(S)36 b Fk(of)31 b(no)s(des)e(w)m(e)j(use) e Fi(\016)s Fk(\()p Fi(S)5 b Fk(\))32 b(to)f(denote)g(the)g(set)g(of)g (edges)g(ha)m(ving)g(exactly)g(one)71 1851 y(endp)s(oin)m(t)e(in)g Fi(S)5 b Fk(.)454 2017 y(minimize)1040 1949 y Fd(P)1136 2044 y Fe(e)1188 2017 y Fi(c)1227 2031 y Fe(e)1265 2017 y Fi(x)1317 2031 y Fe(e)454 2133 y Fk(sub)g(ject)31 b(to)944 2065 y Fd(P)1040 2160 y Fe(e)p Fc(2)p Fe(\016)r Fa(\()p Fe(v)r Fa(\))1265 2133 y Fi(x)1317 2147 y Fe(e)1437 2133 y Fk(=)82 b(2)270 b(for)30 b(ev)m(ery)h Fi(v)d Fh(2)d Fi(V)934 2190 y Fd(P)1030 2285 y Fe(e)p Fc(2)p Fe(\016)r Fa(\()p Fe(S)t Fa(\))1265 2258 y Fi(x)1317 2272 y Fe(e)1437 2258 y Fh(\025)82 b Fk(2)270 b(for)30 b(ev)m(ery)h Fi(S)f Fh(\032)25 b Fi(V)51 b Fk(with)29 b Fi(S)h Fh(6)p Fk(=)25 b Fh(;)31 b Fk(and)e Fi(S)i Fh(6)p Fk(=)25 b Fi(V)1265 2383 y(x)1317 2397 y Fe(e)1442 2383 y Fh(2)87 b(f)5 b Fk(0)p Fi(;)15 b Fk(1)5 b Fh(g)85 b Fk(for)30 b(ev)m(ery)h Fi(e)25 b Fh(2)g Fi(E)162 2547 y Fk(W)-8 b(e)23 b(call)f(the)g(\014rst) f(set)i(of)g(constrain)m(ts)f(the)g(degree)h(constrain)m(ts)f(and)g (the)g(second)h(set)f(the)h(cut)f(constrain)m(ts.)71 2663 y(W)-8 b(e)31 b(need)g(to)g(argue)f(that)h(the)g(ILP)f(ab)s(o)m(v) m(e)h(captures)g(the)f(T)-8 b(ra)m(v)m(eling)31 b(Salesman)e(Problem.) 162 2780 y(Consider)34 b(an)m(y)i(tour)f Fi(T)49 b Fk(and)35 b(put)g Fi(x)1434 2794 y Fe(e)1505 2780 y Fk(=)e(1)j(i\013)f Fi(e)g Fh(2)e Fi(T)13 b Fk(.)57 b(A)36 b(tour)f(con)m(tains)h(exactly)g (t)m(w)m(o)h(edges)g(inciden)m(t)71 2896 y(to)32 b(ev)m(ery)g(no)s(de)e (and)h(hence)g(the)g(degree)h(constrain)m(ts)g(are)f(satis\014ed.)42 b(Also,)32 b(a)f(tour)g(con)m(tains)h(at)g(least)f(t)m(w)m(o)71 3012 y(edges)g(in)e(ev)m(ery)i(cut)g(and)e(hence)i(the)f(cut)h (constrain)m(ts)f(are)h(satis\014ed.)162 3128 y(Con)m(v)m(ersely)-8 b(,)42 b(consider)c(an)m(y)i(solution)e(to)i(the)f(ILP)g(ab)s(o)m(v)m (e)h(and)f(let)g Fi(T)53 b Fk(=)40 b Fh(f)5 b Fi(e)31 b Fk(;)f Fi(y)3056 3142 y Fe(e)3118 3128 y Fk(=)25 b(1)5 b Fh(g)p Fk(.)68 b(The)39 b(degree)71 3245 y(constrain)m(ts)c(ensure)g (that)h Fi(T)49 b Fk(con)m(tains)35 b(t)m(w)m(o)i(edges)f(inciden)m(t)e (to)j(ev)m(ery)f(v)m(ertex)h(and)e(hence)g Fi(T)49 b Fk(is)34 b(a)i(set)g(of)71 3361 y(cycles)28 b(in)e Fi(G)p Fk(.)40 b(W)-8 b(e)29 b(need)e(to)h(sho)m(w)f(that)h(the)g(edges)g(in)e Fi(T)40 b Fk(form)27 b(a)h(single)e(cycle.)41 b(Assume)27 b(otherwise)g(and)f(let)71 3477 y Fi(S)33 b Fk(b)s(e)28 b(the)h(set)g(of)f(no)s(des)g(in)f(one)i(of)g(the)f(cycles.)41 b(Then)27 b Fh(;)f(6)p Fk(=)f Fi(S)30 b Fh(6)p Fk(=)25 b Fi(V)49 b Fk(and)2614 3409 y Fd(P)2710 3504 y Fe(e)p Fc(2)p Fe(\016)r Fa(\()p Fe(S)t Fa(\))2944 3477 y Fi(x)2996 3491 y Fe(e)3058 3477 y Fk(=)25 b(0,)30 b(a)e(con)m(tradiction)71 3593 y(to)j(the)g(fact)g(that)g(the)f(cut)h(constrain)m(t)f(for)h Fi(S)k Fk(is)29 b(satis\014ed.)162 3710 y(The)f(n)m(um)m(b)s(er)g(of)h (cut)g(constrain)m(ts)h(is)e(\(2)1567 3677 y Fe(n)1632 3710 y Fh(\000)17 b Fk(2\))p Fi(=)p Fk(2;)31 b(observ)m(e)f(that)g(the) f(constrain)m(ts)g(for)g Fi(S)34 b Fk(and)28 b Fi(V)38 b Fh(n)18 b Fi(S)34 b Fk(are)71 3826 y(iden)m(tical.)71 3967 y Fm(Miracle:)51 b Fk(The)35 b(T)-8 b(ra)m(v)m(eling)35 b(Salesman)f(ILP)h(can)g(b)s(e)g(solv)m(ed)g(exactly)h(for)f(graphs)g (with)f(up)g(to)i(thousand)71 4083 y(no)s(des.)k(It)30 b(can)h(b)s(e)f(solv)m(ed)g(within)e(a)i(factor)i(of)e(t)m(w)m(o)i(of)f (optimal)e(for)h(v)m(ery)h(large)f Fi(n)p Fk(.)162 4249 y(The)h(linear)e(programming)h(relaxation)h(of)h(an)f(in)m(teger)g (linear)f(program)h(is)f(obtained)h(b)m(y)g(dropping)e(the)71 4365 y(in)m(tegralit)m(y)h(constrain)m(ts.)41 b(F)-8 b(or)31 b(the)f(TSP)-8 b(,)30 b(w)m(e)h(obtain:)547 4531 y(minimize)1133 4463 y Fd(P)1229 4558 y Fe(e)1281 4531 y Fi(c)1320 4545 y Fe(e)1358 4531 y Fi(x)1410 4545 y Fe(e)547 4647 y Fk(sub)5 b(ject)30 b(to)1037 4579 y Fd(P)1133 4674 y Fe(e)p Fc(2)p Fe(\016)r Fa(\()p Fe(v)r Fa(\))1358 4647 y Fi(x)1410 4661 y Fe(e)1529 4647 y Fk(=)83 b(2)h(for)30 b(ev)m(ery)h Fi(v)e Fh(2)24 b Fi(V)1027 4704 y Fd(P)1123 4799 y Fe(e)p Fc(2)p Fe(\016)r Fa(\()p Fe(S)t Fa(\))1358 4772 y Fi(x)1410 4786 y Fe(e)1529 4772 y Fh(\025)83 b Fk(2)h(for)30 b(ev)m(ery)h Fi(S)f Fh(\032)25 b Fi(V)51 b Fk(with)29 b Fi(S)h Fh(6)p Fk(=)25 b Fh(;)31 b Fk(and)e Fi(S)i Fh(6)p Fk(=)25 b Fi(V)1358 4897 y(x)1410 4911 y Fe(e)1529 4897 y Fh(\024)83 b Fk(1)h(for)30 b(ev)m(ery)h Fi(e)26 b Fh(2)e Fi(E)1358 5013 y(x)1410 5027 y Fe(e)1529 5013 y Fh(\025)83 b Fk(0)h(for)30 b(ev)m(ery)h Fi(e)26 b Fh(2)e Fi(E)71 5183 y Fm(Miracle:)62 b Fk(The)41 b(linear)f(program)g (ab)s(o)m(v)m(e)j(can)e(b)s(e)g(solv)m(ed)g(in)e(p)s(olynomial)g(time)i (in)f Fi(n)g Fk(\(=)h(the)h(n)m(um)m(b)s(er)71 5299 y(of)h(no)s(des)f (of)h Fi(G)p Fk(\))g(and)g Fi(L)j Fk(=)1134 5231 y Fd(P)1230 5326 y Fe(e)1282 5299 y Fk(log)r(\(1)29 b(+)f Fh(j)p Fk(\()p Fh(j)p Fi(c)1732 5313 y Fe(e)1770 5299 y Fk(\)\).)79 b(This)41 b(assumes)i(that)g(the)g(edge)g(costs)h(are)f(in)m(tegral.)71 5415 y(Observ)m(e)27 b(that)h(the)g(LP)f(has)h(an)f(exp)s(onen)m(tial)g (n)m(um)m(b)s(er)f(of)i(constrain)m(ts.)40 b(The)27 b(miracle)f(is)h(a) h(consequence)g(of)71 5531 y(the)35 b(Ellipsoid)30 b(metho)s(d)k(for)g (linear)f(programming.)52 b(The)34 b(ellipsoid)d(metho)s(d)j(do)s(es)g (not)g(need)h(an)f(explicit)71 5647 y(description)j(of)i(a)h(linear)d (program.)67 b(The)39 b(metho)s(d)g(w)m(orks)g(in)f(rounds.)65 b(In)38 b(eac)m(h)j(round)c(it)i(pro)s(duces)f(a)p eop %%Page: 6 6 6 5 bop -118 -214 a Ff(6)3321 b(In)n(tro)r(duction)-118 101 y Fk(candidate)31 b(solution)e Fi(x)693 68 y Fc(\003)764 101 y Fk(and)h(asks)h(the)h(question)e(whether)g Fi(x)2058 68 y Fc(\003)2129 101 y Fk(is)g(feasible,)g(i.e.,)i(satis\014es)e(all)g (constrain)m(ts.)-118 217 y(If)g Fi(x)25 184 y Fc(\003)94 217 y Fk(is)g(not)g(feasible,)g(the)g(en)m(vironmen)m(t)g(m)m(ust)h (generate)h(a)e(constrain)m(t)h(violated)f(b)m(y)g Fi(x)2990 184 y Fc(\003)3029 217 y Fk(.)41 b(The)30 b(miracle)f(is)-118 333 y(that)i(the)f(ellipsoid)d(metho)s(d)j(w)m(orks)h(in)e(a)h(p)s (olynomial)e(n)m(um)m(b)s(er)h(of)i(rounds.)-27 500 y(The)j(cut)h (constrain)m(ts)f(can)h(also)g(b)s(e)f(form)m(ulated)g(in)f(so-called)i (inner)d(form.)53 b(Use)35 b Fi(\015)5 b Fk(\()p Fi(S)g Fk(\))36 b(to)f(denote)g(the)-118 616 y(set)d(of)g(edges)h(ha)m(ving)e (b)s(oth)g(endp)s(oin)m(ts)f(in)h Fi(S)5 b Fk(.)45 b(In)31 b(the)h(con)m(text)i(of)e(the)g(degree)g(constrain)m(ts,)h(w)m(e)f(ma)m (y)g(also)-118 732 y(use)633 820 y Fd(X)588 1021 y Fe(e)p Fc(2)p Fe(\015)t Fa(\()p Fe(S)t Fa(\))824 906 y Fi(x)876 920 y Fe(e)938 906 y Fh(\024)25 b(j)p Fi(S)5 b Fh(j)21 b(\000)f Fk(1)91 b(for)31 b(ev)m(ery)g Fi(S)f Fh(\032)25 b Fi(V)50 b Fk(with)29 b Fi(S)i Fh(6)p Fk(=)25 b Fh(;)30 b Fk(and)g Fi(S)g Fh(6)p Fk(=)25 b Fi(V)-118 1177 y Fk(This)32 b(leads)i(to)h(the)f(follo)m(wing)f(form)m(ulation)g(of)h(the)g(T)-8 b(ra)m(v)m(eling)34 b(Salesman)g(Problem)f(where)g(w)m(e)i(also)f(add) -118 1293 y(the)c(redundan)m(t)g(constrain)m(t)g(that)h(a)g(tour)f(con) m(tains)h(exactly)g Fi(n)f Fk(edges.)244 1557 y(minimize)836 1489 y Fd(P)932 1584 y Fe(e)984 1557 y Fi(c)1023 1571 y Fe(e)1061 1557 y Fi(x)1113 1571 y Fe(e)244 1673 y Fk(sub)5 b(ject)30 b(to)740 1605 y Fd(P)836 1700 y Fe(e)p Fc(2)p Fe(\016)r Fa(\()p Fe(v)r Fa(\))1061 1673 y Fi(x)1113 1687 y Fe(e)1232 1673 y Fk(=)83 b(2)306 b(for)30 b(ev)m(ery)i Fi(v)c Fh(2)d Fi(V)724 1730 y Fd(P)820 1825 y Fe(e)p Fc(2)p Fe(\015)t Fa(\()p Fe(S)t Fa(\))1061 1798 y Fi(x)1113 1812 y Fe(e)1232 1798 y Fh(\024)83 b(j)p Fi(S)5 b Fh(j)21 b(\000)f Fk(1)83 b(for)30 b(ev)m(ery)i Fi(S)e Fh(\032)25 b Fi(V)50 b Fk(with)29 b Fi(S)i Fh(6)p Fk(=)25 b Fh(;)30 b Fk(and)g Fi(S)g Fh(6)p Fk(=)25 b Fi(V)810 1855 y Fd(P)906 1950 y Fe(e)p Fc(2)p Fe(E)1061 1923 y Fi(x)1113 1937 y Fe(e)1232 1923 y Fk(=)83 b Fi(n)1061 2039 y(x)1113 2053 y Fe(e)1238 2039 y Fh(2)k(f)5 b Fk(0)p Fi(;)15 b Fk(1)5 b Fh(g)121 b Fk(for)30 b(ev)m(ery)i Fi(e)25 b Fh(2)g Fi(E)-27 2203 y Fk(TODO:)30 b(rewrite)f(the)i(paragraph)f(and)g (use)g(only)f(the)i(inner)d(form)i(of)h(the)f(cut)h(constrain)m(t.)-118 2427 y Fm(The)i(Held-Karp)g(Lo)m(w)m(er)h(Bound)g(for)g(the)g(T)-9 b(ra)m(v)m(eling)34 b(Salesman)f(Problem:)45 b Fk(W)-8 b(e)31 b(do)e(something)-118 2544 y(crazy)-8 b(.)57 b(F)-8 b(or)36 b(ev)m(ery)g(no)s(de)e Fi(v)39 b Fk(w)m(e)d(c)m(ho)s(ose)g(an)f (arbitrary)f(real)h(v)-5 b(alue)35 b Fi(p)2308 2558 y Fe(v)2382 2544 y Fh(2)e Fi(I)-23 b(R)36 b Fk(and)f(de\014ne)f(a)i(mo)s (di\014ed)d(cost)-118 2660 y(function)i Fi(c)283 2674 y Fe(p)360 2660 y Fk(as)i(follo)m(ws.)59 b(F)-8 b(or)37 b(an)g(edge)g Fi(e)f Fk(=)f Fi(uv)40 b Fk(let)d Fi(c)1843 2674 y Fe(p)1883 2660 y Fk(\()p Fi(e)p Fk(\))g(=)e Fi(c)p Fk(\()p Fi(e)p Fk(\))26 b(+)e Fi(p)p Fk(\()p Fi(v)s Fk(\))h(+)f Fi(p)p Fk(\()p Fi(u)p Fk(\).)61 b(The)36 b(v)-5 b(alues)36 b Fi(p)3463 2674 y Fe(v)3540 2660 y Fk(are)-118 2776 y(usually)28 b(called)i(no)s(de)f(p)s(oten)m(tials.)40 b(Clearly)-8 b(,)30 b(for)g(an)m(y)h(tour)f Fi(T)1094 2864 y Fd(X)1094 3061 y Fe(e)p Fc(2)p Fe(T)1240 2951 y Fi(c)1279 2965 y Fe(p)1319 2951 y Fk(\()p Fi(e)p Fk(\))21 b Fh(\000)f Fk(2)h Fh(\001)1659 2864 y Fd(X)1654 3061 y Fe(v)r Fc(2)p Fe(V)1810 2951 y Fi(p)p Fk(\()p Fi(v)s Fk(\))26 b(=)2095 2864 y Fd(X)2095 3061 y Fe(e)p Fc(2)p Fe(T)2242 2951 y Fi(c)p Fk(\()p Fi(e)p Fk(\))31 b Fi(:)-118 3209 y Fk(and)f(hence)g(\()p Fi(v)390 3223 y Fa(0)460 3209 y Fk(is)g(a)h(\014xed)e(but)h(arbitrary)f(v)m(ertex)j(of)e Fi(G)p Fk(\))h(the)g(follo)m(wing)e(linear)f(program)242 3379 y(minimize)722 3311 y Fd(P)818 3406 y Fe(e)870 3379 y Fi(c)909 3393 y Fe(p)949 3379 y Fk(\()p Fi(e)p Fk(\))p Fi(x)1113 3393 y Fe(e)242 3496 y Fk(sub)5 b(ject)30 b(to)742 3428 y Fd(P)838 3523 y Fe(e)p Fc(2)p Fe(\016)r Fa(\()p Fe(v)r Fa(\))1062 3496 y Fi(x)1114 3510 y Fe(e)1234 3496 y Fk(=)83 b(2)306 b(for)30 b Fi(v)f Fk(=)c Fi(v)2091 3510 y Fa(0)725 3552 y Fd(P)821 3647 y Fe(e)p Fc(2)p Fe(\015)t Fa(\()p Fe(S)t Fa(\))1062 3621 y Fi(x)1114 3635 y Fe(e)1234 3621 y Fh(\024)83 b(j)p Fi(S)5 b Fh(j)21 b(\000)e Fk(1)84 b(for)30 b(ev)m(ery)h Fi(S)f Fh(\032)25 b Fi(V)51 b Fk(with)29 b Fi(S)h Fh(6)p Fk(=)25 b Fh(;)31 b Fk(and)f Fi(S)g Fh(6)p Fk(=)25 b Fi(V)812 3677 y Fd(P)908 3772 y Fe(e)p Fc(2)p Fe(E)1062 3745 y Fi(x)1114 3759 y Fe(e)1234 3745 y Fk(=)83 b Fi(n)1062 3862 y(x)1114 3876 y Fe(e)1234 3862 y Fh(\024)g Fk(1)306 b(for)30 b(ev)m(ery)h Fi(e)26 b Fh(2)f Fi(E)1062 3978 y(x)1114 3992 y Fe(e)1234 3978 y Fh(\025)83 b Fk(0)306 b(for)30 b(ev)m(ery)h Fi(e)26 b Fh(2)f Fi(E)-118 4142 y Fk(pro)m(vides)33 b(us)g(with)g(a)i(lo)m(w)m (er)f(b)s(ound)e Fl(LB)1312 4156 y Fe(p)1386 4142 y Fk(for)h(the)i (cost)g(of)f(an)g(optimal)g(T)-8 b(ra)m(v)m(eling)34 b(Salesman)f(tour.)52 b(The)-118 4258 y(lo)m(w)m(er)31 b(b)s(ound)d(dep)s(ends)g(on)j(the)f(p)s(oten)m(tial)g(function)f Fi(p)p Fk(.)-118 4400 y Fm(Miracle:)39 b Fk(F)-8 b(or)27 b(ev)m(ery)g(\014xed)f Fi(p)p Fk(,)h(the)g(v)-5 b(alue)26 b Fl(LB)1518 4414 y Fe(p)1584 4400 y Fk(can)h(b)s(e)f(computed)g(in)f (time)h Fi(O)s Fk(\()p Fi(m)15 b Fk(log)i Fi(m)p Fk(\))26 b(b)m(y)h(\(essen)m(tially\))-118 4516 y(a)k(minim)m(um)c(spanning)i (tree)i(computation;)f Fi(m)g Fk(is)g(the)g(n)m(um)m(b)s(er)f(of)i (edges)g(of)f Fi(G)p Fk(.)-27 4682 y Fl(LB)103 4696 y Fe(p)175 4682 y Fk(is)g(a)j(lo)m(w)m(er)f(b)s(ound)d(on)j(the)g(cost)h (of)f(the)g(optimal)e(tour)i(for)g(ev)m(ery)g(p)s(oten)m(tial)g (function)e(and)h(hence)-118 4798 y(w)m(e)g(ma)m(y)g(consider)1441 4972 y Fl(LB)k Fk(=)24 b(max)1759 5027 y Fe(p)1877 4972 y Fl(LB)2007 4986 y Fe(p)2077 4972 y Fi(:)-118 5179 y Fk(Of)30 b(course,)g Fl(LB)40 b Fk(is)30 b(a)g(lo)m(w)m(er)h(b)s(ound)d (on)i(the)h(cost)g(of)g(an)f(optimal)g(tour.)-118 5320 y Fm(Miracle:)65 b Fl(LB)52 b Fk(is)41 b(equal)h(to)h(the)g(cost)h(of)e (an)g(optimal)g(tour.)77 b(This)40 b(is)i(an)g(example)g(of)h(the)g (so-called)-118 5436 y(Lagrange)31 b(dualit)m(y)-8 b(.)p eop %%Page: 7 7 7 6 bop 1011 458 a Fn(Bibliograph)-6 b(y)1938 6011 y Fk(7)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF
Proof looks basically OK. It needs a justification of how a Turing machine can simulate M in O(f(n)³); "it is safe to say" is a bit weak. The fact that f is time-constructible need to be used explicitly in the proof. Also, we need an article on time-constructible function explaining why the concept is important and the exciting things you can do with functions that are not time-constructible. Gdr 23:41, 2004 Jul 4 (UTC)
- I kind of agree with you that "it is a bit weak" to just say "it is safe to say", but I also believe this article should stick to the thing it is about. To prove that it is possible to simulate a Turing machine in ever-lower time bounds is another proof in itself and, quite frankly, doesn't belong here. Of course, one could remove the "it is safe to say" bit and just claim it's possible in and give the external link; but this runs the risk of the link breaking in future. One could always write a new article with this proof, but the title of that article would be huge... Proof that a Turing machine can simulate the first n steps of another Turing machine in at most n log m time, where m is the length of the description of the second Turing machine and its input perhaps? :) – Timwi 15:51, 3 October 2005 (UTC)
How about Universal Turing Machine? - User:Ben Standeven as 70.249.214.16 22:39, 9 October 2005 (UTC)
Stronger bound, link to proof
The time heirarchy theorem has actually been proved for a much stricter bound. Specifically, TIME(o( f(n)/log(f(n)) )) is a strict subset of TIME(O(f(n)) -- sorry for the lack of pretty LaTeX. Lecture notes with proof here. I'm sorry I don't have time to update the article myself, but I thought I'd at least drop a comment with the information and a link.
Where's the linear lower bound come from?
I often see this result stated with a linear (n) lower bound on t(n). I can only think it must be because no smaller functions are time-constructible (a Turing machine can't even read an input of size n in sublinear time, although it can read it in sublinear work tape space), so it's actually redundant, in the same way that requiring NP proof signatures to have polynomial size is redundant. It might still be useful to mention somewhere though, since it does effectively limit the power of the theorem. Deco 28 June 2005 16:59 (UTC)
- I've never thought of this, but it sounds really interesting. Thanks for this, I'll add something to this effect to the article. – Timwi 15:51, 3 October 2005 (UTC)
Contradiction Cases and Time vs Steps
The cases here:
- If N accepts [N] (which we know it does in at most f(n) operations), this means that K rejects ([N], [N]), so ([N], [N]) is not in Hf, and thus N does not accept [N] in f(n) steps. Contradiction!
- If N rejects [N] (which we know it does in at most f(n) operations), this means that K accepts ([N], [N]), so ([N], [N]) is in Hf, and thus N does accept [N] in f(n) steps. Contradiction!
are not written very clearly. The justification for each step is 'implicit' and hides a mistake in the proof. It would be clearer to write something closer to
0) If N accepts [N] 1) K rejects (N,N) // by definition of N 2) not (N runs on [N] in <= f(n) steps and N accepts [N]) // by definition of K 3) N runs on [N] in > f(n) steps // by 0) and 2) 4) N is in Time(f(n)) // by runtime evaluation of N 3 and 4 are a contradiction.
0) If N rejects [N] 1) K accepts (N,N) // by definition of N 2) (N runs on [N] in <= f(n) steps) and (N accepts [N]) // by definition of K 3) N accepts [N] // simplification of 2) 0 and 3 are a contradiction.
Unfortunately, writing it this way shows a clear mistake. In the accepting case, (N halts on N in > f(n) steps) and (Time(N)=f(n)) does not simplify to false. Since K_f cannot be altered to determine if it's input (M,x) runs in O(f(m)), it is necessary to show that N_f can be defined to run in <= f(n) steps for the contradiction to hold. Antares5245 (talk) 20:33, 20 February 2009 (UTC)
- This error is predicated on the assumption that the notation TIME(f(n)) includes implied big-O notation (TIME(O(f(n))) which I've seen in some definitions but I think in this case it's meant to imply an exact number of steps.
- If the number of steps is exact though, that raises new issues; we can see that K runs in time f(n), but how does N run in time f(n)? If it simulates K, there would be a slowdown. And it has the overhead of copying the input to produce the input for K, and inverting the result, which presumably can't be done in zero time. It seems a constant factor needs to be accounted for somehow, but I'm pretty fuzzy on this. Can someone lay out the details of this part more pedantically? Dcoetzee 21:00, 20 February 2009 (UTC)
- That's phrased a lot better than what I said. Actually more than a constant factor is required if you have to copy the input. That's why the proof seems to imply big O notation, because the added 'constant factors' are ignored. It might actually be necessary for the proof for N to be an alterned 'copy' of K, not just a machine that simulates it.
Textbook-like proof
This article is a great example of why proofs are problematic on Wikipedia. First, it's not clear whether or not this proof is original research. Second, it appears to be a simplified proof, in an attempt to teach the subject matter, rather than present facts. --Taeshadow (talk) 16:52, 11 March 2009 (UTC) +shownstay!