Jo·urnt;e annuelle 2004, p. 5\J 74
POLYHEDRAL COMBINATORICS AND COMBINATORIAL OPTIMIZATION by Alexander Schrijver
1. Introduction Combinatorial optimization searches for an optimum object in a finite collection of objects. Typically, the collection has a concise representation (like a graph). while the number of objects is huge -- more precisely. grows exponentially in the size of the representation (like all matchings or all Hamiltonian circuits). So scanning all objects one by one and selecting the best one is not an option. I\Iore efficient methods should be found. In the l 9GUs. Edmonds advocated the idea to call a method efficient if its rum1iug time is bounded by a polynomial in the size of the represen tation. Since then, this criterion has won broad acceptance. also because Edmonds found polynomial-time algorithms for several important com binatorial optimization problems (like the matching problem). The class of polynomial-time solvable problems is denoted by P. Further relief in the landscape of combinatorial optimization v.ras dis covered around 1970 when Cook and Karp found out that several other prominent combinatorial optimization problems (including the traveling salesman problem) are the hardest in a large natural class of problems. the class N'P. The class ::-JP includes most combinatorial optimization problems. Any problem in NP can be reduced to such ·NP-complete· problems. All NP-complete problems are equivalent in the seuse that the polynomial-time solvability of one of them implies the same for all of them.
© Journee annuelle. SJ\IF 2004 ALEXANDER SCHRLlVER 60
Almost every combinatorial optimization problem has since been either prowd to lw polyuomial-time soh·able or NP-complete - and none of the problems have been proved to be both. This spotlights the big mystery: are the two properties disjoint (equivalently, P:fNP), or do they coincide (P=NP)? Polyhedral and linear programming techniques have turned out to be essential in sol-\'iug combinatorial optimization problems and studying their complexity. Often a polynomial-time algorithm yields, as a by product. a description (in terms of inequalities) of an associated poly hedron. Conversely. an appropriate description of the polyhedron often implies the polynomial-time solvability of the associated optimization problem. by applying linear programming techniques. \Vith the duality theorem of linear programming. pol~·hedral characterizations yield min max relations. aud vice versa. This area of discrete mathematics is called polyhedral combinatorics. \\'e give some basic. illustrative examples. For an extensive suIYey, we refer to Schrijver [39]. Background on linear programming can be found in [38].
2. Perfect matchings Let G = (F. E) be an undirected graph. A perfect matching in G is a set ,\! of disjoint edges covering all vertices. Let 11• : E ---. R+. For any perfect matchiug .\/. denote
(1) u·(M) := L u·(e). t:E;\f
\Ve \Nill call w ( i\I) the weight of JI. Suppose iww that \Ve want to find a perfect matching J\J in G v.;ith weight w ( 1U) as small Rs possible. In notation. •ve want to 'solve'
( 2) min { w( ll I) I M perfect matching in G}.
This problem shows up in sewral practical situation. for instance ·when an optinmm assignment or schedule has to be determined. \\"e can formulate problem (2) equivalentl_\· as follov."s. For any perfect matching .U. denote the incidence vector of JI in JRE by x-' 1; that is.
JOURNEE ANNlJELLE POLYHEDRAL COl\IBI'.'JATORICS AND Cm.IBINATORIAL OPTI:\!IZATION 61
if e E ]\[. (3) if c ~ M. for c E E. Considering w as a vector in JRE. we have 1c(JJ) = u·T \.H. Hence problem (2) can be re-written as .
(4) min{u,T XM I Al perfect matching in G}.
This amounts to minimizing the linear function 1cr.r over a finite set of vectors. Therefore. the optimum value does not change if we minimize over the rnr1ve:r h:nll of these vectors:
(5) rnin{u·T:r I :r E conv.hull{ \.\!IM perfect matching iu G} }.
The set
(G) conv .lrnll {XM I M perfect matching in G} is a polytope in IRE, called the perfect matching polytopr of C:. As it is a polytope. there exist a matrix A and a \·ector b such that
(7) conv.hull{\M I M perfect matching in G} = A.r ::=; b}.
Then problem (5) is equivalent to
(8) min{u·T:r I A.r :Sb}.
In this wav we havt' formulated the original combiuaturial as a linea·;· programrnfrlg problem. This enables u:; tu pl\1- 0 Tannnino methods to stuclv the original probkm. . b o . . l 11 JW to find t lw w•tlF 1 .rnd The question at this point is. iowever. 1 · · . . · . "' · .. . l d I ! . exist l mt \\'t' nm~! kilt ·w· t I1<•rn the vector b. \\e know t iat .-1 an Jc o · · , ' ·' · in order to apply linear prognunming metlluds._ , ,, ,. 1 < · · . I·· 1. d I "Ul eas1h· tie :.\ ...; ... p.1, For hi.partde graphs, sue 1
bipartite. the matching polytope of c; is equal to the set of all \"ectors :r E JPI,£ S<1tist\·ing
(9) :r(e) 2:: 0 for ( E £. L:c(c) = 1 for 1· E l '.
(The sum ranges over all edges e containing c.) This is in fact equirnlent to a theorem of Birkhoff [2]. saying that each doubl)· stochastic matrix is a couwx combination of pennutatiull matrices. (A matrix is doubly 8tochastic if it is 1101111egatiw mid each row sum aud each column sum is equal to 1. A pcrnrntatiun matri.r is a 0. l matrix with pre('isely one 1 in Pach nw; and each column.) It is not difficult to slim\· that the perfect matching pol)iope for bi partite grnphs is indeed complett>ly determirn'd hy ( 9). First uotf' that the perfect rn Theorem 1. The incidence matri1· Ac; of a hipartih graph G = ( \ ·. E) i8 totally unimodular. Proof. Let B he a square submatrix of Al~· of ordPr k :,;ay. \\'e sliuw that clet B equals 0 or ±1 by induction on t. If k = L tlw statemt>nt is triYial. So let k > 1. \\"e distinguish three cases. Ca:::1: 1: B ho::: o column with 0111,11 o·,,_ Tlwn det B=O. Case :l: B has a culwn'n 1cith e1:11ctl11 on1 1. lH thar <'dS<' we call write (possibly after permuting rows or culunms ): 1 1/ ) ( rn) B = ( O B' . for sonw matrix B' and yector b. when• 0 dt•nure~ th~· all-zew Yedor in IR.1- 1• BY the induction 1 . ln-potlwsis.. dt·t B' oi: {U.::::l}. Heme. !n-: li • det BE {Cl. ±1}. POLYHEDRAL COMBINATORICS AND COMBINATORIAL OPTI~!IZAT!ON 63 Case 3. Each col'Urnn of B contains e.ractly two 1 ·s. Then. since G is bipartite. we can write (possibly after permuting rnws): in such a way that each column of B' contains exactlv one 1 and each column of B" contains exactly one 1. So adding up all r~ws in B' gives the all-cme vector, and also adding up all rmvs in B" gives the all-one vector. The rnws of B therefore are linearly dependent. and hence det B=O. I The total unirnodularity of Ac implies that the vertices of the polytope determined by (9) are integer vectors. i.e .. belong to 1'.E. Now each integer vector satisfying (9) must trivially be equal to y;\/ for some perfect matching ,\!. Hence. (12) if G is bipartite. the perfect matching polytope is determined by (9). \\'e therefore can appl~' linear programming techniques to handle prob lem ( 2). Thus \Ve can find a minimum-weight perfect matching in a bipar tite grflph in polynomial time, with an:v polynomial-time linear program ming algorithm. l\Ioreover. the duality theorem of linear programming gives (13) min{w(M) I J\J perfect matching in G} = min{u•Tx I :r?:: O.Ac1· = 1} = max{yTl I y E lR".yTA.c?:: u,T}. ( 1 denotes an all-one vector.) This is an example of a min-rna:r fonnula. that can be derived from a polyhedral characterization. Conversely. min max formulas (in particular in a weighted form) often give polyhedral characterizations. The polyhedral description together with linear programming duality also gives a certificate of optimality of a perfect matching :\I: to c·om·ince your 'boss· that a certain perfect matching J 1 has minimum \veight. it is J)()Ssible and sufficient to display a vector .lJ in ~ 1, satisfying y T.4.c; ?:: U'T and yTl = w(1\l). In other \vords. it yields a good characterization for the minimum-weight perfect matching problem in bipartite graphs. SuCIETE ~!ATHE~!AT!ql'E DE FHANCE 2004 ALEXANDEH SCHRI.JVEH 64 3. But what about nonbipartite graphs? For general. nonbipartite graphs G. the perfect matching polytope is not determined by (9). For instance. if G is an odd circuit. then the vector :1.· E ~E defined by ;r(e) := ~ for all e E £. satisfies (9) but does not belong to the perfect matching polytope of G (as G has no perfect matching at all). A pioneering and central theorem in polyhedral combinatorics of Ed monds [8] gives a complete description of the inequalities needed to de scribe the perfect matching polyiope for arbitrar~· graphs: one should add to ( 9) the inequalities ( 14) L :t( e) 2': 1 for each odd-size subset (' of F. fEo(U) Here 5 ( U) denotes the set of edges ccmnecting c· and \ · \ C. Trivially. the incidence vector \ M of an~· perfect ma tchiug M satisfies (14). So the perfect matching polytope of(; is contained in the polytope determined b:v ( 9) and ( 14). The content of Edmonds· theorem is the converse inclusion. Theorem 2. Fo·r any gn1ph G. the per:fi:d rtw.tching pol.l}topc is deter rn:ined by (9) and (14). Proof. Clearly. the perfect matching pol:dop<' is contained in tlw pol~· tope Q determined b)· (9) and (14). Suppose that the converse indusiou does not hold. Then we can choose a wrtex .r of Q that i::J not in the perfect uwtching polytope. V..1e mav assume that we haw chosen this counterexample such that 11/I + IEI. is as small as possible. Hence U < .l'(e) < 1 for all f E E (otherwise. if :r( e) = 0. we can delete e. and if .r( e) = 1. we can delete e a,nd its ends). So each degree of G is M least 2. and hence IEI 2': WJ. If IEI = 11/I. each degree is 2. in which case the theorem is trivialh· true. So 1£1 > lvl Note also that IFI is ewn. since otherwise Q = 0 (consider U := \ 1 in ( 1-±)). As .r is a vertex of Q, there exist IEI linearly iudepeudeut constraints among (9) and (1.J) satisfied with equality. Since IEI > Jl'I. there is an odd subset U of 1/ with 3 '.S ICI '.S WI - 3 aml L,"',1 11 · 1 .r(f) = l. .JOURNEE ANNUELLE POLYHEDRAL COMBINATORICS AND COl\IBINATORIAL OPTIMIZATION 65 Consider the projections x' and x" of x to the edge sets of the graphs G/U and G/U. respectively (where U := V \ U, and where G/H' is the graph obtained from G by contracting all vertices in H' to one vertex). Here we keep parallel edges. Then x' and x" satisfy (9) and (14) for G/U and G/U, respectively, and hence belong to the perfect matching polytopes of G /U and G /U, by the minimality of IVI + IEI. So there is a k such that G /U has perfect matchings l\f{, ... , f\I[ and G / U has perfect matchings AI{', ... , AI{: with k k I 1 U' II 1 AI" (15) X = - L x· I and :r = - L x I • k k i=l i=l (Note that :.r is rational as it is a vertex of Q.) Now for each e E 6(U). the number of i with e E M; is equal to k:J:'(e) = kx(e) = k:r"(e), which is equal to the number of i with e E Mf'. Hence i.ve can assume that, for each i = 1 ..... k, MI and Mf' have an edge in 6(U) in common. So M; :=MIU Mf' is a perfect matching of G. Then k l (Hi) .r = - "'""' \.\I,. kL- i=l Heuce .r belo11gs to thf' perfect matching polytope of G. I In fact. Edmonds designed a pol:n10mial-time algorithm to find a min imum-weight perfect matching in a graph. ·which gave this polyhedral characterization as a l>:v-product. Conversely, from the characterization one may derive the polynomial-time solvability of the weighted perfect matching problem. Iu applying linear programming methods for this. one will be faced with the fact that (9).(14) consists of exponentially many inequalities. since there exist exponentially many odd-size subsets U of V. So in order to solve the problem with linear programming methods. we cannot just list all inequalities. However. the ellipsoid method for linear programming (Khachiyan [26]) does not require that all inequalities are listed a priori ([22,23]). It suffices to have a polynomial-time algorithm ansi.vering the question: S()C!ETE ~IATHEt-.IATHlUE DE FllANCE 2011-1 ALEXANDER SCHRI.JVER 66 ( 17) given x E JRE. does :.r belong to the perfect matching poly- tope of G? Such an algorithm indeed exists. as it has been shown that the inequalities (9) and (14) can be checked in time bounded b~· a polynomial in J\'I. IEI. and the size of x. This method obviously should avoid testing all inequalities ( 14) one by one. Combining the description of the perfect matching polytope \Yith the duality theorem of linear programming gives a min-max formula for the minimum weight of a perfect matching. It again yields a certificate of optimality: if we have a perfect matching 1\l. we can convince our ·boss' that M has minimum weight. by supplying a dual solution y of objectiw value w(1\l). So the minimum-weight perfect matching problem has a good characterization - i.e .. belongs to NPnco-NP. This gives one motivation for studying polyhedral methods. The el lipsoid method proves polynomial-time soh·ability. it however does not yield a practica1 method. but rather an incentiw to search for a practi cally efficient algorithm. The pol~'hedral method can be helpful also in this, e.g., by imitating the simplex method with a constraint generation technique. or by a primal-dual approach. \Ve note that Edmonds' theorem is equirnlent to the following. (18) The convex hull of the symm.etr·ic permutation matrices in :JR"x" is equal to the set of doubly stochastic matrices with the property that for each odd number k and each principal submatrix B of order k. the sum of the entries in B is at most k - l. 4. Hamiltonian circuits and the traveling salesman problem As we smv, perfect matchings form an area ·where the search for an inequality system determining the corresponding polytope has been suc cessful. This is in contrast with. for instance. Hamiltonian circuits. (A H a.rniltonian cirrnit is a circuit covering all vertices.) No full descrip tion in terms of inequalities of the convex hull of the incideuce wctors of edge sets of Hamiltonian circuits - the tra:ueling salesman polytope - is JOURNEE ANNUELLE POLYHEDRAL CO'.IIBINATORICS AND CO:'IIBIJ:\ATORJ..\L OPTii\IIZ~\TIO:\ 67 knmvn. Tlw corresponding optimiwtion problem is the trawling sales man problem: 'find a Hamiltonian circuit of minimum weight". \\·hid1 problem is ::\P-complete. This implies that. unless NP=co-:\P. there exist facet-i11duci11g in equalities for the tnweling salesman pol:.iope that haw no polynomial time certifiC'ate of valiclit~·. Otherwise. linear programming duality \\·oulcl yield a good characterization. So unless ::\P=co-NP there is no hope for an appropriate characterizatio11 of the trawling salesman pol.\iope. It can be seen that the following ·obvious· set of inequalities is not enough to determine the traveliug salesman pol~iope: (19) :r(c) 2: U fore EE. :L::r(e)=2 for t' E I·. c31• L .r(r) 2: 2 for 0 "I l' "I\'. 'E In matrix terms. unless I\P=co-'.'JP. there is no hope for an appropriate description of the convex hull of those n x n permutation matrices made by a pennuation ·with precisely one orbit. (Simply requiring that the entries in any nonempty proper principal submatrix of order k add up to at most k - 1 is not Pnough.) l\Ioreowr. u11less I\P=P. there is no polynomial-tiulf' algorithm an- svvering the question (20) given .r E JRE. does .r belong to the traye[ing salesman poly- t opC'·? Otherwise. the ellipsoid method would giw the polynomial-time solvabil ity of the traveling salesman problem. Nevertheless, pol~·hedral combinatorics can be ~llOETE ~Ic\THE~L.\Tllll"F DE F!L\'.'\CE ~1111-1 ALEXANDER SCHRIJVER 68 polytope, the better the bound is. This can be verY useful in a branch and-bound algorithm. This idea originates from Da~1tzig. Fulkerson. and Johnson [6]. · 5. Stable sets and semidefinite programming Related to the problems described abm·e is the problem of finding a maximum-size stable set in a graph G = (F. E) (and more generally. a maximum-\veight stable set. but we will restrict ourselYes here to the cardinality case). Here a subset S of V is called stable if any two wrtices in Sare nonadjacent in G. The problem comes up in practice for instance when assigning frquencies to radio stations or mobile phones. Finding a maximum-size stable set is again an NP-complete problem. so no good description of the corresponding stable set pol.11tupe (the conwx hull of the incidence vectors in JR.'. of the stable sets) ma~· be expected. However. for certain graphs. the so-called perfect graphs. a maximum size stable set can be found in polynomial time ( [22]). The basic idea is to apply semidefinite programming to calculate the following bound of Lovasz [3'1] on the maximum size a( G) of a stable set in G: (21) rJ(G) := max{lT.~Jl I J/ E JR.nx" positin' semidefinite. Mi,j = 0 if ij E E. trace( JI) = 1}. Here we assume without loss of generality that G has ,·ertex set {1. .... n}. .. To see that a(G) ~ rJ(G). let S be a maximum-size stable set in G. and define the matrix JI by where xs is the incidence vector of S in JRI'. taken as column rector. Then 1\1 is positive semidefinite. J/1.J = 0 if ij E £. and trace.\/ = l. So o(G) = ISI = 1TM1 ~ iJ(G). . . The valued( G) can be calculated in polynomial time. as it is a :~cm~def- inde programming problem. A generic form of sueh a pro~lem is: .g1Yeu c1 •...• c1 E JR. and real symmetric matrices Ao ..... A1. B \of t>qual d11.11~·n sions). find 1·1 .... . J.'t E JR. that maximize L; C;.l'; subject to the nmd1t1011 JOURNEE ANNl'ELLE POLYHEDRAL COl\IBINATORICS AND COl\IBIN:\TORl:\L OPTI:\IIZ:\TIO'.\" 69 that (Li. :i·iA.i) - B is positive semidefinite. If all A; and B are diago nal ma~nces. we have a linear programming problem. Semidefinite pro grammmg problems can be solved in polynomial time. with the ellipsoid method or with an 'interior-point method" . . It follows that if a(C) = iJ(G). we can calculate n(G) in pol~·nomial time. l\foreover. if (23) a(C') = rJ(C') for each induced subgraph G' of G. then we can find a ma..ximum-size stable set in pol~·nomial time. ( A.n ·induced subgraph is a subgraph (\/'. £') of (l ·. E) with \ ., ~ \ · and P = {ij EE I i,j E F'}.) \Vhich graphs satisfy (23)? First. it was shown b~· Lonisz [35] that these are precisely the graphs G such that (24) a(C') = -y(C') for each induced subgraph C' of G. Here 'y(H) denotes the colouring number of H. and H denotes the com plementary graph of H (·whose edges are precisely the non edges of HI. Berge [l] introduced the name perfect for graphs G satis~·ing ( 24 i. and he conjectured that these are precisely those graphs G with t lit' property that neither G nor G contains a chordless circuit uf odd length ~ 5. (Necessity of this condition is easy.) This strong prrftef graph conjecture was only recently prowd by Chudnovsky. Robertson. Se~·muur. and Thomas [3]. requiring deep decomposition techniques for ~mph:". Berge's weak perfect graph conject·ure. stating that the complement (,' uf any perfect graph is perfect again. was shown earlier b~· Lov(lsZ [:33]. 6. Historically The first min-max relations in combinatorial optimization wert' pni\·(•d by Denes Konig [27,28]. on edge-colouring and i~iatchings it~ bipartitt> graphs. and by Karl l\lenger [36]. 011 disjoint paths m graphs. 1 ht'ma,tch ing theorem of Konig was extended to the weighted n1st> h:\' EgerYarY : 1-1, · The proofs by Konig and Egervar~· were in prindp~~l alg_oritlnuic .. ,anLl ~1:su for l\Ienger's theorem an algorithmic proof was g1w11 m the Hl.JUs. I he theorem of Egervary may be seen as polyhedral. ALEXANDER SCHRIJ\'ER 70 Applying linear programming techniques to combinatorial optimiza tion problems came along with the introduction of linear programming in the 1940s and 1950s. In fact. linear programming forms the hinge in the history of combinatorial optimization. Its initial conception by Kantorovich [25] and Koopmans [29] was motivated by combinatorial applications, in particular in transportation and transshipment. After the formulation of linear programming as generic problem, and the development in 194 7 by Dantzig [5] of the simplex method as a tool, one has tried to attack about all combinatorial optimization problems with linear progTanuning techniques, quite often very successfully. In the 1950s. Dantzig [4]. Ford and Fulkerson [16,15.17]. Hoffman [24]. Kuhn [30.31]. and others studied problems like the transportation. ma.ximum flow, and assignment problems. These problems can be reduced to linear programming by the total unimodularity of the underlying matrix. thus yielding extensions and polyhedral and algorithmic interpretations of the earlier results of Konig. Egervary. and l\lenger. Kuhn realized that the polyhedral methods of Egervary for \Veighted bipartite matching are in fact algorithmic. and yield the efficient ·Hungarian· method for the as signment problem. Dantzig, Fulkerson, and Johnson [6,7] gave a solution method for the traveling salesman problem, based on linear programming with a rudimentary. combinatorial version of a cutting plane technique. A considerable extension and deepening. and a major justification, of the field of polyhedral combinatorics was obtained in the 1960s and 1970s by the work and pioneering vision of Edmonds· [8.9.10,11,12.13]. He characterized basic polyiopes like the perfect matching polytope. the arborescence polytope. and the matroid intersection polytope: he intro duced (with Giles) the important concept of total dual integrality: and he advocated the interconnections between polyhedra, min-ma.x relations, good characterizations, and efficient algorithms. Also during the 1960s and 1970s. Fulkerson [18.19,20.21] designed the clarifying framework of blocking and antiblocking polyhedra. throwing new light by the classical polarity of vertices and facets of polyhedra on combinatorial min-max relations and enabling, with a theorem of Lehman [32], the deduction of one polyhedral characterization from another. It stood at the basis of the solution of Berge 's weak perfect graph conjec ture in 1972 by Lovasz [33]. As mentioned, Berge's strong perfect graph conjecture was recently proved by Chudnovsky. Robertson. Seymour, and Thomas [3] . .JOURNEE ANNUELLE POLYHEDRAL CO!dBINATORICS AND COJ\IBINATORIAL OPTI.MIZATION 71 References [l] C. Berge. Some classes of perfect graphs. in: Si:i: Papers on Graph The or:I/ [related to a series of lPCtures at the Research aud Training School of the Indian Statistical Institute. Calcutta, l\farch-April 1963]. Research and Training School, Indian Statistical Institute, Calcutta. [1963,] pp. 1-21. [2] G. Birkhoff. 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