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Ideal Binary Clutters, Connectivity and a Conjecture of Seymour

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Ideal Binary Clutters, Connectivity and a Conjecture of Seymour IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR GERARD´ CORNUEJOLS´ AND BERTRAND GUENIN ABSTRACT. A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that ¼; ½ intersect with odd cardinality a fixed given subset of elements. Let A denote the matrix whose rows are the Ü ¼ : AÜ ½g characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron f is ×Ø Ì Ì integral. Examples of ideal binary clutters are ×Ø-paths, -cuts, -joins or -cuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour conjectured that a binary clutter is ideal if and only if it does not µ Ä Ç b´Ç à à contain F , ,or as a minor. In this paper, we show that a binary clutter is ideal if it does not contain 7 5 5 five specified minors, namely the three above minors plus two others. This generalizes Guenin’s characterization Ì of weakly bipartite graphs, as well as the theorem of Edmonds and Johnson on Ì -joins and -cuts. 1. INTRODUCTION E ´Àµ A clutter À is a finite family of sets, over some finite ground set , with the property that no set of jE ´Àµj À fÜ ¾ Ê : À contains, or is equal to, another set of . A clutter is said to be ideal if the polyhedron · È Ü ½; 8Ë ¾Àg ¼; ½ i is an integral polyhedron, that is, all its extreme points have coordinates. A i¾Ë À = ; À = f;g À A´Àµ clutter À is trivial if or . Given a nontrivial clutter , we write for a 0,1 matrix whose ´Àµ Ë ¾À columns are indexed by E and whose rows are the characteristic vectors of the sets . With this fÜ ¼ : A´ÀµÜ ½g notation, a nontrivial clutter À is ideal if and only if is an integral polyhedron. Ì E ´Àµ À Ì À Given a clutter À, a set is a transversal of if intersects all the members of . The clutter ¡ ´Àµ À E b´Àµ = E ´Àµ b´Àµ b , called the blocker of , is defined as follows: and is the set of inclusion-wise ¡ b b´Àµ = À À;b´Àµ minimal transversals of À. It is well known that [13]. Hence we say that form a blocking pair of clutters. Lehman [14] showed that, if a clutter is ideal, then so is its blocker. A clutter is said Ë ;Ë ;Ë ¾À Ë 4 Ë 4 Ë ¾ ¿ ½ ¾ ¿ to be binary if, for any ½ , their symmetric difference contains, or is equal to, a set of À. i ¾ E ´Àµ À=i ÀÒi Given a clutter À and , the contraction and deletion are clutters defined as follows: ´À=iµ=E ´ÀÒiµ= E ´Àµ fig À=i E , the family is the set of inclusion-wise minimal members of Ë fig : Ë ¾Àg ÀÒi = fË : i 6¾ Ë ¾Àg f , and . Contractions and deletions can be performed sequentially, À  and the result does not depend on the order. A clutter obtained from by a set of deletions d and a set of   \  = ; À ÀÒ = c d d c contractions c , (where ) is called a minor of and is denoted by .Itisaproper minor  [  6= ; d if c . A clutter is said to be minimally nonideal (mni) if it is not ideal but all its proper minors are ideal. Date: March 2000, revised December 2001, March 2002. Key words and phrases. Ideal clutter, signed matroid, multicommodity flow, weakly bipartite graph, Ì -cut, Seymour’s conjecture. Classification: 90C10, 90C27, 52B40. This work supported in part by NSF grants DMI-0098427, DMI-9802773, DMS-9509581, ONR grant N00014-9710196, and DMS 96-32032. 1 2GERARD´ CORNUEJOLS´ AND BERTRAND GUENIN µ Ç E ´Ç ½¼ Ã Ç Ã 5 à The clutter à is defined as follows: is the set of edges of the complete graph and is 5 5 5 à 5 ½¼ the set of odd circuits of 5 (the triangles and the circuits of length ). The constraints corresponding to the ½ ½ ½ ´ ; ;:::; µ fÜ ¼ : A´Ç µÜ triangles define a fractional extreme point of the associated polyhedron à 5 ¿ ¿ ¿ ½g Ç Ä F . Thus à is not ideal and neither is its blocker. The clutter is the family of circuits of length three 5 7 E ´Ä µ=f½; ¾; ¿; 4; 5; 6; 7g of the Fano matroid (or, equivalently, the family of lines of the Fano plane), i.e. F 7 and ¨ © Ä = f½; ¿; 5g; f½; 4; 6g; f¾; ¿; 6g; f¾; 4; 5g; f½; ¾; 7g; f¿; 4; 7g; f5; 6; 7g : F 7 ½ ½ ½ ´ ; ;:::; µ Ä The fractional point is an extreme point of the associated polyhedron, hence F is not ideal. 7 ¿ ¿ ¿ Ä Ä F The blocker of F is itself. The following excluded minor characterization is predicted. 7 7 Seymour’s Conjecture [Seymour [23] p. 200, [26] (9.2), (11.2)] Ä Ç b´Ç µ à à A binary clutter is ideal if and only if it has no F ,no , and no minor. 7 5 5 · · Ø 6¾ E ´Àµ À E ´À µ= Consider a clutter À and an arbitrary element . We write for the clutter with · À = fË [fØg : Ë ¾Àg É E ´É µ E ´Àµ [fØg 6 and . The clutter 6 is defined as follows: is the set of edges Ã É Ã É 6 4 7 of the complete graph 4 and is the set of triangles of . The clutter is defined as follows: ¾ ¿ ½ ¼ ½ ¼ ½ ¼ ½ ½ ¼ ¼ ½ ¼ ½ ½ 6 7 ¼ ½ ¼ ½ ½ ¼ ½ 6 7 6 7 ¼ ½ ½ ¼ ¼ ½ ½ : A´É µ= 7 6 7 ½ ½ ¼ ¼ ¼ ¼ ¼ 4 5 ¼ ¼ ½ ½ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ½ ½ ¼ ¡ A´É µ A b´É µ 6 Note that the first six columns of 7 form the matrix . The main result of this paper is that Seymour’s Conjecture holds for the class of clutters that do not have · · É É and minors. 6 7 · · Ä Ç b´Ç µ É É Ã Ã Theorem 1.1. A binary clutter is ideal if it does not have F , , , or as a minor. 7 5 5 6 7 Since the blocker of an ideal binary clutter is also a ideal, we can restate Theorem 1.1 as follows. · · µ µ Ä Ç b´Ç b´É b´É µ à à Corollary 1.2. A binary clutter is ideal if it does not have F , , , ,or as a minor. 7 5 5 7 6 G E ´Àµ G À We say that À is the clutter of odd circuits of a graph if is the set of edges of and the set of odd circuits of G. A graph is said to be weakly bipartite if the clutter of its odd circuits is ideal. This class of graphs has a nice excluded minor characterization. Ç Theorem 1.3 (Guenin [10]). A graph is weakly bipartite if and only if its clutter of odd circuits has no à 5 minor. The class of clutters of odd circuits is closed under minor taking (Remark 8.2). Moreover, one can easily Ç check that à is the only clutter of odd circuits among the five excluded minors of Theorem 1.1 (see 5 Remark 8.3 and [20]). It follows that Theorem 1.1 implies Theorem 1.3. It does not provide a new proof of Theorem 1.3 however, as we shall use Theorem 1.3 to prove Theorem 1.1. Ì Ì Consider a graph G and a subset of its vertices of even cardinality. A -join is an inclusion-wise Ì G[ ] minimal set of edges  such that is the set of vertices of odd degree of the edge-induced subgraph . IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 3 Æ ´Í µ:=f´Ù; Ú µ:Ù ¾ Í; Ú 6¾ Í g Í A Ì -cut is an inclusion-wise minimal set of edges , where is a set of jÍ \ Ì j Ì Ì vertices of G that satisfies odd. -joins and -cuts generalize many interesting special cases. If = f×; Øg Ì Ì ×Ø ×Ø G Ì , then the -joins (resp. -cuts) are the -paths (resp. inclusion-wise minimal -cuts) of .If = Î Ì jÎ j=¾ G Ì Ì , then the -joins of size are the perfect matchings of . The case where is identical to the Ì set of odd-degree vertices of G is known as the Chinese postman problem [6, 12]. The families of -joins and Ì -cuts form a blocking pair of clutters. Ì Theorem 1.4 (Edmonds and Johnson [6]). The clutters of Ì -cuts and -joins are ideal. The class of clutters of Ì -cuts is closed under minor taking (Remark 8.2). Moreover, it is not hard to check that none of the five excluded minors of Theorem 1.1 are clutters of Ì -cuts (see Remark 8.3 and [20]). Thus Ì Theorem 1.1 implies that the clutter of Ì -cuts is ideal, and thus that its blocker, the clutter of -joins, is ideal. Hence Theorem 1.1 implies Theorem 1.4. However, we shall also rely on this result to prove Theorem 1.1. The paper is organized as follows. Section 2 considers representations of binary clutters in terms of signed matroids and matroid ports. Section 3 reviews the notions of lifts and sources, which are families of binary clutters associated to a given binary matroid [20, 29]. Connections between multicommodity flows and ideal clutters are discussed in Section 4. The material presented in Sections 2, 3 and 4 is not all new. We present it here for the sake of completeness and in order to have a unified framework for the remainder of the paper.
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