On Ideal Clutters, Metrics and Multiflows

Beth Novick 1 and AndrOs Seb6 ~

I Clemson University, Clemson, South Carolina 29634-1907, USA 2 CNRS, Laboratoire Leibniz, Grenoble, France

Abs t rac t . "Binary clutters" contain various objects studied in Combi- natorial Optimization, such as paths, Chinese Postman Tours, multiflows and one-sided circuits on surfaces. Minimax theorems about these can be generalized in terms of ideal binary clutters. Seymour has conjectured a characterization of these, and the goal of the present work is to study this conjecture in terms of multiflows in matroids. Seymour's conjecture is equivalent to the following: Let yr be a binary clutter. Then the Cut Condition is su~cient in the underlying matroid, for all F E :F as demand.set, to have a muitiflow, if and only if it implies the so called Ks, Fr and Rio.conditions. These three conditions are applications of the general "Metric Condi- tion" to particular 0 - 1 bipartite weightings. In this paper we prove the following weakening of this conjecture: The Cut Condition is sufficient for all F E Jr as demand-set, to have a multiflow, ff and only flit implies the Metric Condition [or every bipartite 0 - 1 weighting. A special case of this result has been stated as a conjecture in Robert- son and Seymour's "Graph Minors" volume, (1991, "Open Problem 11 (A. SebS)'). Using Lehman's theorem on minimal non-ideal clutters, we sharpen the properties of minimally non-ideal clutters for the binary and graphic special cases.

1 Introduct ion

A clutter is a family of bts of a finite ground set N, none of which contains any other. We will suppose that every e E N is contained in at least one set of the family. A clutter 7/ is ideal (has the max-flow-re_in-cut property) if its blocking polyhedron, that is the polyhedron {z E IR~. : z ( n ) ___ 1 for all a E 7/) , has only integer vertices. Clearly, a clutter is ideal precisely when the set of vertices of its blocking polyhedron are exactly the characteristic vectors of its blocker. The blocking clutter or blocker of the clutter .4 C_ 2 N, denoted by B = b(7/), is defined to be the family of minimal elements of {B C N : IB n A I > 1 for all A E `4}. It is easy to see b(b(`4)) = `4, [7]. The notation `4 and B will automatically mean that these are clutters and they are the blocker of each other. We denote the ground set of a specific clutter `4 by N(.4). When it causes no confusion, we will speak interchangeably about a subset of N and its incidence vector considered as a member of GF(2) n, where n := INI; similarly, a family of subsets and a 0-1 matr ix are interchangeable, as well as the mod 2 sum of vectors and their "symmetric difference", denoted by A. dPe linear independence, rank, span,

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orthogonality etc. is usually understood over over GF(2) n, when we understand it over the reals we will mention it. The sign "-" means congruence modulo 2.

A binary clutter is the family of (inclusionwise) minimal supports of elements of affine subspaces (shifts of linear subspaces) of vector spaces over GF(2). For results on binary spaces and binary clutters see [17], [21], [3], [4]. It is easy to see that the blocker of a binary clutter 7/ is b(7/) = {K C_ N : tK f) H I - 1 mod 2 for all H E 7/, K minimal ). It follows that b(7/) is also a binary clutter (see [17]). The result of deleting or contracting e E N in a binary clutter 7 / is denoted by n\e, and 7//e respectively, and defined by 7/\e := { g E 7/ : e ~ H), and 7//e := the minimal elements of {H - {e) : H E 7t}. 7 / \X /Y denotes the the result of deleting the elements of X and then contracting those of Y and is called a minor. It is easy to see that b(7/\ v) = b(7~)/v and that the classes of ideal and binary clutters are minor-closed. Throughout the sequel clutter will mean "binary clutter."

Binary clutters generalize various objects studied in combinatorial optimiza- tion such as paths, Chinese postman tours and one-sided circuits on surfaces. Seymour, in his works [21] and [20] shows how good characterization theorems about these special cases can be extended to binary clutters, and conjectures the following. ~7 will denote the clutter consisting of the lines of Fang plane;/C5 the set of odd circuits of the complete graph/(5.

Seymour~s Con jec tu r e : A binary dotter is ideal i[ and only if it contains none of lCb, b(1Cb) or Jr7 as minors.

Compare the three excluded minors of this conjecture to the infinite set of minimal non-ideal clutters (see some in 'hire classes in I6]~ [14], [12]).

The collection of all odd circuits of an undirected graph and its blocker are binary clutters. More generally, a signed graph is the pair (G, R), where G = (V, E) is an undirected graph, and R, C_ E(G). We define the pair of binary clutters

A(G, R): = {A: IA 0 R[ is odd, A a circuit of G}.

B(G, R) : = {B = RAQ : Q is a cocycle of G, B minimal non-empty}.

A(G, R) is called an odd circuit clutter, and B(G, R), the blocker of A(G, R), a signing clutter. The class of odd circuits clutters of graphs is minor closed. Signed graphs are studied by Gerards in [3], .[4], where the various particular cases have a topological meaning. The approazh of the present work is different: we see the idealness of binary clutters as a problem about metrics and multiflows in matroids; exploiting this tool we get a characterization of idealness which sharpens Lehman's theorem (also using it) for binary clutters.

It is straightforward to extend the definition of signed graph to signed ma- troid: a pair (M, R), where R C_ N(M). In this context, A(M, R) (B(M, R)) is A(G, R) (B(G, R)) with G replaced by M in the above definition and is, again, called an odd circuit clutter (a signing clutter). It is dear that R E B(M, R). The generalization will be useful in the context of multiflows, since every binary clutter is an odd circuit clutter for some M and some R~ see (2.4).

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The core of a binary clutter ,4 is the family of its minimal cardinality ele- ments; the family will be denoted by core (A), and the cardinality of its elements by r := r(A); s := s(A) := r(b(A)). Let MA (and MB) be a matr ix whose rows are the incidence vectors of the members of core (,4) (core (b(A)) respectively). The n x n identity matr ix will be denoted by I, whereas J is the n x n matr ix of all l 's. We state Lehman's theorem on minimal non-ideal clutters, making obvious simplifications due to the binary case (the degenerate projective plane is not a binary clutter) :

T h e o r e m 1.1 [ L e h m a n [22]] If`4 is a minimalnon-ideal binary clutter, then the unique non-integer vertex of its blocking polyhedron is the constant vector l / r , rs > n + 1,and M.4, MB are n x n matrices whose row and column sums are r and s respectively, and the rows of these matrices can be ordered so that M A M ~ = J + ( r s - n) I = MXB M~. []

The key statement is that the row and column sums are r and s respec- tively. As Seymour [22] observes, this provides sufficient information for a coNP- characterization of idealness. Still, the matr ix equations of Theorem 1.1 also express combinatorial facts that will be important for us. Let us look more into these for later use. Let a = rs - n + 1. By Theorem 1.1 each A E core (,4) is paired to a unique B e core (B) with [A N B[ = a. The sets A and B will be called partners of each other.

Besides the equation M.~Mg = J + (rs - n ) I which provides us with the cardinalities of the intersections we have M g M.4 = J + (rs - n)I , expressing the following:

(1.1) [22] Let By := {B E core(B) : the partner o r B contains v}. Then for all v E N, By~v, is a partition of N \ v. Out of the r members of Bt, exactly a contain v.

If ,4 and B are blockers of each other and the corresponding matrices M~t, MB have row and column sums equal to r and s respectively and satisfy the equations M~4M~ = J + ( r s - n ) I - MgM.~, then we say that the clutters are partitionable.

A partitionable binary clutter is not ideal, and it is also not necessarily minimal non-ideal. It is easy to see that the binary clutter ,4' generated by the core of ,4 (that is, X consists of the set of minimal supports of vectors obtained by summing an odd number of rows of A), is also partitionable. It will be called the normalization of `4. If .4' = `4, then `4 will be said to be normalized. It is easy to see that by normalization, the core, r, s, and the core of the blocker do not change; the normalization is the minimal binary clutter for which the core is the same as originally.

The fact that M~ and M g commute can be summarized in a simple "po- larity" relation between the core of a minimal non-ideal clutter and the set of elements {1 , . . . , n}. An analogous notion for antiblocking (minimal imperfect graphs), defined by Tucker [23], has also been useful.

We define the polar of the partitionable binary clutter `4 to be the family of (inclusionwise) minimal supports of vectors in the affine space generated by the columns of A.

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(1.2) The polar A* of a partitionable binary clutter is also a binary clutter which is partitionable. The binary clutter ,4** is the normalization of A. The normalization of b(A* ) is B*. t3

The proof is straightforward from Theorem 1.1. The analogous equality for "antiblocking" plays an important role in Gasparian's recent proof [2] of Lov~z's perfect graph theorem.

Note that the three known minimal non-ideal clutters are normalized and self-polar.

In Section 2 we prove some basic relations between binary clutters, met- rics and multiflows in underlying matroids, which are used in the following two sections to prove the main results. Section 3 provides a coNP characterization for the Cut Condition to be sufficient for a class of multiflow problems whose demand-edges are members of a binary clutter, and Section 4 sharpens this char- acterization. The property we characterize is a reformulation of idealness, and is interesting for its own sake. It sharpens the application of Lehman's theorem to binary clutters, using Theorem 1.1 and the above mentioned consequences. In Section 5 we make some additional comments about the graphic special case of our results and the most important graphic special case of Seymour's conjecture.

2 C l u t t e r s ~ M e t r i c s a n d M u l t i f l o w s

Prior to presenting our main results, we will need to introduce some matroidal tools and to clarify some powerful connections between ideal binary clutters, on the one hand, and multiflows and metrics in matroids, on the other. Such is the topic of the present section.

For us, matroid wilt mean "binary matroid" - - one representable over GF(2) and will be a pair M = (N,C), where C is the set of its circuits. The linear

space generated by C is called the cycle-space of M. (Its members are the cycles; the circuits are the inclusionwise minimal cycles.) The linear space orthogonal to C is called the cocycle space, its elements are the cocycles, or cuts, its (inclusion- wise) minimal nonempty elements are the cocircuits. The matroid M* = (N, C*)~ where C* is the set of cocireuits of M, is the dual matroid of M. If the columns of a matrix represent a binary matroid, then the rows generate the cocycle space of the matroid. The ground-set N of a matroid M will be referred to as N(M) . The rank function of the matroid is denoted by r := r(M). For more on the basic notions and simple facts related to binary clutters and matroids, see the introductions of [17], [21], [19], [20], [31, [4]~ T h r e e M a t r o i d s Assoc ia ted wi th each B ina ry C l u t t e r (see more details in [151)

Let 7 /be a binary clutter. We define the down matroid of T/ to be M0(7/) := (N, Co), where Co consists of the minimal non-empty subsets of N that can be written as the sum of an even number of elements of 7/. The up matroid of 7/ is defined by M1(7/) := (N, C1), where Cl is the set of minimal non-empty elements of the linear space generated by 7/. Proofs of (2.t), (2.2) and (2.3), below, are straightforward and hence omitted (see also [15]).

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(2.1) If # 7/ r {0}, then Co generat a s u p p l e of corank - r(G) -

r(C0) = 1 of t1 , and 7 / = el\Co. D One gets M1 from M0 by "undeleting" and contracting an element, and the

matroid one gets in the intermediate step is uniquely determined:

(2.2) I f (~ r 7 / r {~}, there ex/sts a uniquely determined matroid M2, and t e S(M2) such that 3/[2(,4)\t = Mo(A), M 2 ( A ) / t - MI(A). []

Ms will be called the port matroid of 7/. Given our assumption that ~ r 7/ r {~}, M0, M1, M~ are uniquely deter-

mined by 7/. a Conversely, the pair (M0, 3//1) or the pair (M2, t), t e N(M2) uniquely determines 7/.

(2.3) Let A C 2 N be a binary clutter, and let B be its blocker. For e E N, i f A\e r $ and A/e r {~}, then M~(A\e) - Mi(A) \e and Mi(A/e) = Mi(`4)/e, (i - 0, 1,2). Moreover, the matroids Mi(`4), M~(B) (i = 0, 1,2) relate as follows:

(i) Ms(A) \ t = Mo(`4), / t = (ii) M2(B) = M; (,4). (iii) Mo (B) = M~ (`4).

[]

We make the convention that the down, up and port matroids of the clutter `4 = ~ is any triple of matroids M0 = (g, C0), 3'/1 = (N, C1), 3//2 = ( g U {t}, C2), such that t is a coloop in M2 and Mx = M2/t. Accordingly, the down, up and port matroid of B = {~}(= b(`4)) is any triple where t is a loop in 3,/2 and Mo = M2\t . It follows given this convention, that the claims of (2.3) hold for the blocking pair of clutters ~ and {~} as well.

The following is not difficult to show:

(2.4) Every binary clutter .4 is equal to `4(M, R) for some uniquely deter- mined binary matroid M, and arbitrary R E b(A); the same is true for B(M, R); `4(M,R) and B(M, R) are the blocker of each other. Furthermore,

(i) I f `4 is an arbitrary binary clutter, then `4 = `4(M,R), B = B(M,R) with M = MI(A), and arbitrary R e b(A);

(ii) `4\e = A ( M \ e , R ) ; `4/e = `4(M/e,R') , where IV e B, e ~ R', or i f e e R' for all R' e B, then `4/e = {~}. []

The fact that the sets `4 = A(M, R), B = B(M, R), and e E N ( M ) do not change if we replace R by any R ~ E B, (that is, by R A D where D is an arbitrary eocycle), is used in proving 2.4 (ii).

Note that a binary clutter `4 is a (graphic) odd circuit clutter if and only if M1 (`4) is graphic. The graph obtained by removing, that is deleting, a set of edges Y from a graph G then shrinking to a single node, that is contracting each of the edges X E E(G), where X N Y = ~, is denoted G / X \ Y . Since the process corresponds to the same minor taking in the graphic matroid,

` 4 ( a / x \ r , R) = (`4(G, R ) ) / X \ Y .

a These statements are equivalent to the well-known fact due to Lehman[10] that a binary clutter uniquely determines the underlying matroid.

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Idea l C l u t t e r s , Mul t i f l ows a n d Me t r i c s . Let M = (E, C) be a matroid. m : E ~-~ lR+ t2 {oo} is called a metric, if for every circuit C E C and every e e C: re(e) < m(C\e). Metrics on M form a cone, whose extreme rays are called primitive metrics. A cut metric is a function 1 on a cocycle, 0 elsewhere; if the cocycle is a cocircuit, it is easy to see that the cut metric is primitive. A function d : E ~-~ IN is called a distance function if for some F C E it is defined in the following way: for e E F, d(e) := 1, and for e ~ F, d(e) := min{[C\e[ : C e C, {e} = C\F}. It is easy to see that a distance function is a metric and that it is finite if and only if N ( M ) \ F does not contain a cocircuit. This metric will be denoted by [M, El. For instance, [M(gs), E(K2,a)] is 1 on a K2,3 subgraph of Ks, and 2 otherwise.

We define now some other basic examples of metrics. Let L be a line of FT; C3 is any 3-element subset of any 4-element circuit of R10; R10 := M0(K:s). All of [FT, F T - L], [Ks, K2,3], [R10, R10 - C3] can be checked to be primitive (follows from (3.1) below).

Besides these, Papernov's multiflow problem (H6, R) is also a useful example serving often as a counterexample: H6 is the graph (unique up to isomorphism) one gets from K5 by uncontracting an edge ab so that no series edges occur; R consists of three edges ab, xlx2 and yly2 forming a matching of Hr and such that a is adjacent to both xl and x2 and b is adjacent to both Yl and Y2. The metric [Hr Hs\R] is primitive.

We define an [M, F]-metric to be a function m on the elements N of an arbitrary matroid such that contractit elements with m-value 0 and deleting some of the e E N with re(e) = m(C\e) for some circuit C, we get the matroid M, with the metric [M, F] (up to isomorphism). Clearly, [M, F]-metrics are metrics. Cut-metrics are [M, g] metrics where M is the matroid having one coloop element. A metric m : N(M) ~-+ Z+ is bipartite if ~eer re(e) is even for all circuits C of M. All the examples above are bipartite metrics. A binary matroid is called bipartite if every circuit is even and Eulerian if every cocircuit is even.

A multiflow problem, (M, R, c), on a matroid M = (g , C) is defined by a set of demands R C N, and a function c : N ~+ l~+. A multiflow is a function f : C ~+ Q+, so that if f(C) > 0, then IC n R[ = 1, and the sum of the f-values of circuits containing a given e E N is at most c(e), moreover equality holds here for e E R. (M, R) will stand for the class of multiflow problems in M with demands R. For basic facts about multiflows in matroids we refer to [20], for their connection to binary clutters see [20] or [4]. For metrics in matroids we refer to [16]o If e e R, c(e) is called the demand of e, if e E N \ R it is called the capacity of e.

The connection between metrics and multiflows is provided by the following statement, well-known for graphs, which is also easy from linear programming duality (Farkas' lemma) for matroids (see [16]).

M e t r i c C r i t e r i o n For R C N(M) and c : N(M) ~ It+ there exists a multittow if and only if the following Metric Condition Js satisfied: for every primitive metric m, E~el~ m(e)e(e) < BeeN(M)\R m(e)c(e). [3

The Metric Condition specialized to a class # of metrics will be called the

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# Condition. The Metric Condition specialized the incidence vectors of cocycles is called Cut Condition (see [20]). The matroid M is called R-flowing ([20]) for R C_ N(M), if for arbitrary c : N(M) ~ R+ for which the Cut Condition is satisfied there exists a multiflow.

Let us illustrate these definitions on / /6 with the above introduced notation: let c be 1 everywhere except on ab, and c(ab) := 2. It is easy to see that the Cut Condition and the [Ks, K2A-Condition are satisfied, but the [H6, Hs\R]- condition is not satisfied. This is the only violated metric condition, and ,4(//6, R) is not minimally non-ideal ! However, the edges adjacent to a form a tight cut, switching on which the following [/{5, K2,s] Condition is violated: m(ab) := 0, re(e) := 1 for e E H6\R' and re(e) := 2 for e E R ~, where R' is the set of demands after the switching.

By the polarity of cones, primitive metrics provide the facets of the "multi- commodity cone":

(2.5) The matroid M is R-flowing (R C_ N(M)), if and only if the Cut Condi- tion implies the m-condition for every primitive metric m, where m(e) = m( C\ e) f o r c E R .

A cocycle for which the Cut Condition holds with equality will be called a tight cocycle. It is easy to see that switching on a tight cocycle (that is inter- changing the edges in R and those which are not in R), we get an equivalent multiflow problem which has an (integer) solution if and only if the original problem has one. If the Cut Condition holds then a tight cocycle is the disjoint union of tight cocircuits, and a sequence of switchings can be replaced by just one, on a tight eocycle. (Note that the emptyset is a tight cocycle.) The example of / /6 (see above) shows that the condition proving the non-existence of a flow may become simpler after switching on a tight cut.

Seymour pointed out the connection between the cut condition and idealness. (for instance in [20]). The following statement includes this connection plus the above remarks about the up, down matroid and primitive metrics. The proof relies only on an understanding of these notions, we omit it here:

(2.6) Let .A be a binary clutter, 13 = b(A). The following statements are equivalent: (i) A is ideal. (ii) M~) (,4) is A-flowing for all A e A. (iii) M1 (A) is B-flowing for all B e 13. (iv) For every multiflow problem (M, R), where M = M1 (,4) and R e B, i f e : N(M) ~-~ Z+ satisfies the Cut Condition, then for every metric it also satisfies the Metric Condition. []

3 T h e S u f f i c i e n c y o f D i s t a n c e F u n c t i o n s

Throughout this section and the next we will suppose that ,4 is minimal non- ideal and binary with b(A) := B, and we will use the notations r, s, core, M~t, M~, introduced in Section 1. If R E core (B), let R = N(.A) - R. Let ! denote the vector, of appropriate dimension, of all l's.

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We next prepare for the main result of this section, Theorem (3.1).

(3.1) I rA is partitionable, then with arbitrary R E core ( B), m := [MI(A),R] is a primitive metric.

Indeed, by definition, m is a metric. Clearly, i fA E core (.4) and AMR : {e), then m satisfies the equality re(e) = m(A \ e) ( : r - 1). But according to Lehman's theorem (Theorem 1.1), out of the n linearly independent vectors A e core (A), n - 1 satisfy IA N RI = 1. Hence for n - 1 linearly indepen- dent A 1 , . . . , An-1 e core (A) (all but one member of core (A)), the inequality m(A \ e) - m(e) > 0 is satisfied with equality, where {e} = A N R. Note that the coefficient vectors of these equalities arise by switching the sign A 1 , . . . , An-1 (from + to - ) for all e E R. But clearly, a set of vectors is linearly independent if and only if after switching the signs of some coordinates it is linearly indepen- dent. Hence m satifies n - 1 linearly independent equations, which means that it is an extreme ray of the cone of metrics. [3

Such a metric will be called partitionable; if ,4 is in addition minimal non- ideal, then we will call the metric minimal non-ideal as well.

The most natural converse of (3.1) is not true, as the example of [H6, R] (see Section 2) shows. Still, with some care, (3.1) can be reversed:

(3.2) Let M = (N, C) be a binary matroid, and R* C g . I f M is not R*- flowing, then there exists R E B(M,R*) (that is, R = R*AC, C cocycle of M, R minimal) and c : N ~-~ I{+ such that for the muRiflow problem (M, R = R* AC, c), there exists a violated minimal non-ideal metric condition.

Proof . Suppose M is not R*-flowing. Then according to (2.6) .4(M, R*) is not ideal, whence there exist X , Y C_ N(M), X f l Y = ~ such that Q : - A(M,R*) \ X / Y , 7~ := b(Q) is a minimally non-ideal pair of clutters. Denote the respective minima of cardinalities by q and r. Denote the cardinality of N(Q) - N(7~) by n. Let R E core (7~) be arbitrary, and C := R*zhR. Define m(e) := 0 if e E Y, re(e) := 1 i f e e N ( M ) \ ( R O X U Y ) ; i f e E R O X , then re(e) := rain{iV\el : C E C, {e} = C N (R U Z)} . Define now c(e) := 0 if e E X, c(e) to be arbitrarily big if e E Y, and c(e) := 1 otherwise. (M, R, c) satisfies the Cut Condition, sincoevery cut C of M, R A C E B(M, R*), so c(RZlC) >_ c(R), that is, 0 < c(RAC) - c(R) = c(C \ R) - c(C N R); but the Metric Condition is violated, since ~ e e R m(c)c(e) = (q - 1)r > n -- r = IN(M) \ (X U Y U R)I = E N(M \R m(e)c(e), o

In the example of [H6,R], C is the tight cut mentioned in Section 2; this ex- ample shows that (3.2) cannot be sharpened so that for (M, R*) itself (and some capacity function) there exists a violated minimal non-ideal metric condition. A special case of the following theorem was stated as a conjecture in Robertson and Seymour's "Graph Minors" volume, (1991, "Open Problem 11 (A. Seb~)").

T h e o r e m 3.1 Let ,4 be a binary clutter, B its blocker. Then MI(A) is B- flowing for MI B E 13, if and only i f for every capacity function for which the cut condition is satisfied, every m-condition where m is a distance function, is also satisaed. Furthermore, m can be restricted to be a minimal non-ideal metric.

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P r o o f . The only if part is easy, using results in Section 2. Indeed, suppose (M1 (.4), B) is B-flowing for all B �9 B . Let c : S ~-~ R + be such that the Cut Condition is satisfied. Then there exists a multiflow, and by (the easy part of) the Metric Criterion, the Metric Condition is satisfied for every metric m, in particular also for distance functions, as claimed.

To prove the if part, suppose B �9 13 and (M1 (.A), B) is not B-flowing. Then by (3.2) there exists R �9 13(= 13(Ml(A),B) so that for some c : S ~ l~+ there exists a violated minimally non-ideal metric condition for (M1 (.A), R, c). []

4 B i p a r t i t e D i s t a n c e F u n c t i o n s

The next result, Theorem 4.1, relies on the application of Lehman's theorem to binary clutters, and not at all on the theory of metrics and multiflows; yet, through Theorem 3.1 it will yield a surprising statement on the primitive metrics of binary matroids.

(4.1) I f A is minimal non-ideal and binary then the cardinalities of M1 dements of A have the same parity.

P r o o f . We prove that 1 is the rood 2 sum of incidence vectors of some B' C core (B). We will be done then, since [A I = xAT1 = Xa y Y~Bet~' XB = ~-,BeB' I A N Bh where IAnBI is odd for all A �9 A. So for all A �9 A, IAI has the same parity as 113'1, as claimed.

We show now using Theorem 1.1 that )"]~BeBv XB, where By is the set defined in (1.1), is 1, so B ~ = By will do. We actually know from (1.1) the exact value of w := ~'~Bet~ XB: w(u) = 1 if u �9 N \ v and w(v) = a. But using Theorem 1.1 again, tr is also equal to IAnB[ for some A �9 .4 and B �9 B (namely for any pair of associates), so it is odd. n

We get as a consequence:

(4.2) Minimal non-ideal metrics are bipartite, that is, i r a is minimal non- ideal and R �9 13, then m := [MI(A),-R] is a bipartite metric.

P r o o f . Let C be a cycle in MI(A). If C is also in Mo(A) then (4.1) implies that ICI is even. Since I R n C I is odd, and hence also I t \ h i is even, it follows that:

m(e) = re(e)+ = IRnCI(8-1)+IC\RI- 0 eEC eERr eEC\FI

Similarly, if C �9 M1 (.A)\M0(A)(--~4), then IRACI is odd; according to (4.1) [C I = s; these two observations together imply ]C\RI = s - 1, and we deduce:

)--:re(e) = I R n C l ( s - 1 ) + l C \ R I = ( s - 1 ) + I C \ R I - 0 . eEC

O

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With the help of (4.2) Theorem 3.1 can be sharpened:

T h e o r e m 4.1 Let ,4 be a binary clutter, B its blocker. The Cut Condition is sufficient for (MI(A), B) (B E B) to have a multiflow, i f and only if for every capacity function for which the cut condition is satisfied, every m-condition where m is a bipartite distance function is also satisfied.

Theorem 4.1 is an immediate consequence of Theorem 3.1 and (4.2). One can actually deduce further structural properties from Lehman's theorem in the particular case of binary clutters:

(4.3) Let Q and n be a minimally non-ideal pair of binary clutters, M := MI(Q), and R e core(R). Let Q be the partner of R and C := Q o R, [C I is odd, [C I >_ 3. Consider the multiflow problem ( U, R, I_). (i) M is an Euleri matroid, either M or M / n (in both cases M - a ) is bipartite. (ii) There exist exactly n - 1 non-empty tight cuts each of which meets C in IC[- 1 elements. (iii) I f K is a tight cut, then it is an (inclusionwise) minimal cut, furthermore K \ R is a minimal cut in M - R.

Proof . By Theorem 1.1 [C[ = a > 1. and a = IQ N R[ is odd, since Q and n are a blocking pair of binary clutters.

By (4.1), every member of n has the same parity, so M0(n) is bipartite, and then M~(n ) - MI(Q) = M is Eulerian. Applying (4.1) to Q now, we get that all members of Q have the same parity. If this parity is even, then M = Mx(Q) is bipartite; if it is odd, then, since [R N Q[ is odd for all Q E Q~ we get that M / R is bipartite. The proof of (i) is finished.

Let now the different minimum cardinality members of ~ be R = R0, R1, �9 �9 Rn-1; according to Theorem 1.1 this is a complete list. Therefore, since B E n is of minimum cardinality if and only if Rz~B is a tight cut, the complete list of tight cuts is RAR~ (i = 1 , . . . , n - 1). Since Q ~ R - C, but [Q N R~[ = 1 if i = 1 , . . . , n - 1, [(RZ~R/) N CI = ICI - [Ri N CI _> IC] - IP~ Cl QI = IcI - 1, and (ii) is proved.

Now (ii) implies that any pair of non-empty tight cuts have IcI - 2 _> common demands, so they are not disjoint. If K is a tight cat which is not minimal, then, since the matroid is binary, it is the disjoint union of at least two tight cuts, which is impossible, because we have just noticed that tight cuts are not disjoint. If K - R is he disjoint union of the cuts C1 , . . . , Ck of M - R, then there exist R I , . . . , R k such that Ci UR/ is a cut (i = 1 , . . . , k ) . Since K is a minimal cut, it follows that R1 U . . . U Rk = R (not necessarily disjoint union). Now

k

0 _< E ( I C i ] - t/~l) _< IK \ R [ - IK f3RI = 0,

so there is equality throughout. It follows that the R~ are disjoint and Ci U Ri (i = 1 , . . . , k) are tight cuts, so by what has already been proved, i = 1, and (iii) is proved. []

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5 Summary

In this section we summarize the results and illustrate them in the graphic special case. We reformulate Seymour's conjecture, putting it into a form which helps compare it with what has been proved. We then show what remains to be proved for showing this conjecture for graphs This special case contains many important multiflow problems in graphs [4], so deserves special attention.

C o n j e c t u r e Let M = (N, C) be a binary matroid, and R C N. Then either M is R-flowing, or there exists R ~ E B(M, R) (that is, R' = R A C , where C is a cut and R ~ minimal with this property) and c : S ~-~ Z so that the multiflow problem ( M, R', c) satisfies the Cut Condition, but some [FT, F7 - L], [Ks, K2,3], [R10, R10 - Ca] condition is violated.

For graphs, only one of these conditions can be violated, the [K5, K2,a] con- dition. The proof of the equivalence of this and Seymour's conjecture is not difficult, we omit it here. (It uses (2.6).) Let us restate Theorem 4.1 using (4.3) in the special case of graphs:

T h e o r e m 5.1 Let G = (V, E) be a graph, and R C. E. Then either G is R-flowing, or there exists a c : E ~-~ 2g (in fact 0 - 1 - o% suppose there are no O-weight edges [contract them]) so that possibly after switching on a cocycle there exists a minimal non-ideal metric m for which the metric condition is not satisfied, but the Cut Condition is satisfied. This metric m has the following properties: Let Q be the partner of R and C := Q N R, [C[ is odd, [C[ _> 3. Consider the multiflow problem (M, R, !). (i) G is an Eulerian graph, G - R is connected and bipartite and either MI de- mand edges join two points in different classes or all of them join two points in the same class. (ii) There exist exactly n - 1 non-empty tight cuts each of which meets C in ] C [ - 1 elements. (iii) I f K is a tight cut, then it is an (inclusionwise) minimal cut, furthermore K \ R is a minimal cut in M - R.

In the graphic case Theorem 3.1 becomes a coNP-characterization theorem where the negative result is particularly easy to check: either the Cut Condition is satisfied for every B E B, or i f not, there exists a B C B and a 0 - 1-weighting of the edges for which the length of the shortest paths violates the metric criterion.

In order to prove Seymour's conjecture for graphs, we need to prove that the only metrics with the properties listed in Theorem 5.1 are [Ks, K2,3]-metrics.

If G is a graph and R C E(G), let us call a path R-geodesic if it occurs as a subset of a shortest path in R between the endpoints of an uv E R.

Consider the union of three disjoint even R geodesics with the same endpoints a and b at distance d. (It follows from Theorem 1.1 and a _> 3 that these geodesics exist.) If the set of pairwise distances between the mid-points cl, c~, Ca is also d, and the minimum paths between ci and cj (i ~ j) are R-geodesics, we will call this disjoint union of R geodesics a kite.

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Let G be a graph and R C E(G). - I f there exists an R-kite, A(G, R) has a K:5-clutter minor. (In (//6, R), see the definition in Section 2, there exists a kite !)

The following also seems to be plausible: - I f there exists no R-kite, for all R' E B(G, R) (including R) the Cut Condition is sufficient to have a flow.

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