Bertrand Guenin - the set covering problem

A family of sets over a finite groundset where no set is properly contained in another is a clutter. Given a clutter \(\mathcal F\) we can define the set packing polytope,

The clutter \(\mathcal F\) is said to be perfect if the associated set packing polytope \(Q(\mathcal F)\) is integral. Padberg, Chvátal, and Lovász proved that a clutter \(\mathcal F\) is perfect if and only if its members are the maximal cliques in a perfect graph. Thus the study of perfect clutters reduces to the study of perfect graphs. The Strong Perfect Graph theorem of Chudnowsky, Robertson, Seymour and Thomas, then easily leads to an excluded minor characterization for perfect clutters.

\[ P(\mathcal F)=\{x\geq{\bf 0}: \sum_{v\in S}x_v\geq 1,\; S\in{\mathcal F}\}. \]

The clutter \(\mathcal F\) is said to be ideal if the associated set covering polyhedron \(P\) is integral. Ideal clutters, in contrast to perfect clutters, are not associated with a single class of combinatorial objects.

Examples of ideal clutters,

Not all examples of ideal clutters arise from graphs, there are some purely geometric constructions as well. Thus ideal clutter form a very rich and complex class of objects.

Problem

Characterize ideal clutters.

A solution to this problem seems to be out of range at this juncture, so we will consider special cases. Here we shall focus on excluded minor characterizations. Let us formalize this. Let \(\mathcal F\) be a clutter and let \(P'\) be the polyhedron obtained from \(P(\mathcal F)\) by setting variables to zero and projecting variables. Then \(P'\) is of the form \(P(\mathcal F’)\) for some clutter \(\mathcal F’.\) We then say that \(\mathcal F'\) is a minor of \(\mathcal F.\) As projection and faces of integral polyhedra are integral, idealness is a minor closed property.

A clutter is said to be minimally non-ideal (mni) if it is not ideal, but every proper minor is. Lehman proved that minimally non-ideal clutters have a unique fractional extreme point, moreover, that fractional point satisfies some very strong regularity properties. However, this is only a partial characterization, and (in contrast to minimally non perfect clutters) there are complicated infinite classes, and lots of sporadic examples of mni clutters. An accessible proof of Lehman's proof is available in the next reference. Results about general mni clutters can be found in the second reference.

To make further headway we need to restrict ourselves to special classes of clutters.

Binary clutters

A signed matroid is a pair \((M,\Sigma)\) where \(M\) is a binary matroid and \(\Sigma\subseteq EM.\) A cycle of \(M\) is even (resp. odd) if \(|C\cap\Sigma|\) is even (resp. odd). Clearly, the set of inclusion-wise minimal odd cycles of \((M,\Sigma)\) is a clutter. Clutters that can be obtained by this construction are said to be binary.

If characterizing all mni clutters is difficult, what about binary mni clutters? At least we have the following beautiful 1977 conjecture by Seymour,

The Flowing Conjecture

The only minimally non-ideal binary clutters are the clutters of

the Lines of the Fano,
the odd circuits of \(K_5\), and
the complement of the cuts of \(K_5.\)

For (1) the groundset of the clutter are the elements of the Fano matroids, and the members the lines. We illustrate the corresponding clutter \(\mathcal F\) in the following picture,

For (2) and (3) the groundset are the edges of the complete graph on \(5\) vertices. Note for (2) the members are the triangles and the pentagons and for (3) the members are triangles together with an independent edge and complete graphs on four vertices. The conjecture remains open but there are lots of known special cases as we will discuss next.

Known cases of the Flowing Conjecture

Consider a binary clutter \(\mathcal F\) and it associated signed matroid \((M,\Sigma)\), recall that \(M\) is a binary matroid. When \(M\) is co-graphic the members of \(\mathcal F\) are precisely the \(T\)-cuts of some graft \((G,T).\) As none of outcomes (1)-(3) of the Flowing Conjecture can be expressed as \(T\)-cuts, the conjecture implies that clutters of \(T\)-cuts are ideals. This is indeed the case and was proved by Edmonds and Johnson. When \(M\) is graphic the members of \(\mathcal F\) are precisely the odd circuits of some signed-graph \((G,\Sigma).\) It is easy to verify that (1) and (3) are not clutters of odd circuits of any signed-graph. Thus the Flowing Conjecture predicts that (2) is the only obstruction to idealness in that case. Indeed this is the case,

Weakly Bipartite Theorem

Clutters of odd circuits of signed-graphs are ideal if and only if it they do not contain as a minor the clutter of odd circuits of \(K_5.\)

An immediate consequence of this result are examples (1) and (2). This has implications for multi-commodity flow problems. For a survey on flows in graphs and matroids see,

Combining the Weakly Bipartite Theorem, the Theorem of Edmonds and Johnson on \(T\)-cuts and Seymour's decomposition of regular matroids, we can obtain sufficient conditions, in term of excluded minors, for a binary clutter to be ideal.

An extension to the Weakly Bipartite Theorem and an extension to the Edmonds and Johnson \(T\)-cuts Theorem is given in

Furthermore, we develop an algorithmic version of Lehman's theorem with applications, including the following,

Theorem

For any graph \(G\) with non-negative weight edges, we can in polynomial time do one of the following, (a) find a maximum weight cut, or (b) find a set of edges \(B\) and a cut \(\delta(U)\) disjoint from \(B\) such that the graph obtained from \(G\) by deleting \(B\) and contracting all edges of \(\delta(U)\) is \(K_5.\)

The Weak Flowing Conjecture

Consider a clutter \(\mathcal F\), the blocker of \(\mathcal F\) is the clutter \(b(\mathcal F)\) with the same groundset as \(\mathcal F\) where the members are the inclusion-wise minimal sets that intersect every member of \(\mathcal F.\) The blocking operation is an involution and Lehman proved that idealness is preserved under taking the blocker. This implies in particular, that if a clutter is mni then so is its blocker. Observe that the blocker of the Lines of the Fano is itself, and that the blocker of the complements of cuts of \(K_5\) is the clutter of odd circuits of \(K_5.\) As the blocker of a binary clutter is also binary, the Flowing Conjecture can be restated as follows,

The Flowing Conjecture - alternate statement

Let \(\mathcal F\) be a minimally non-ideal binary clutter. Then \(\mathcal F\) or \(b(\mathcal F)\) is the clutter of the Lines of the Fano or the clutter of the odd circuits of \(K_5.\)

Observe that both the clutter of the Lines of the Fano and the clutter of odd circuits of \(K_5\) have a member of cardinality three. Hence, a weakening of the Flowing Conjecture is the following,

The Weak Flowing Conjecture

Let \(\mathcal F\) be a minimally non-ideal binary clutter. Then \(\mathcal F\) or \(b(\mathcal F)\) has a member of cardinality three.

We proved that in fact the Weak Flowing Conjecture is equivalent to the Flowing Conjecture.

2. The Cycling Conjecture

A family of sets over a finite groundset where no set is properly contained in another is a clutter. A cover of a clutter is a subset of the groundset that intersects every member of the clutter. The maximum number of pairwise disjoint sets in a clutter is the packing number and the cardinality of the smallest cover is the covering number. Clearly, the packing number is at most the covering number and a clutter packs if both numbers coincide. Not every clutter packs, consider for instance,

Examples of clutters that pack,

We will describe (4) in more details later. Examples (3) and (4) both involves a parity condition. For (3) all circuits have even length and for (4) all edge cuts have an even number of edges. The Cycling conjecture will provide sufficient conditions for a clutter, satisfying a particular parity condition, to pack.

The conjecture,

Consider a clutter \(\mathcal F\) with an element \(e\) of the groundset. The deletion minor \(\mathcal F\setminus e\) is defined as \(\{S\in\mathcal F: e\notin\mathcal F\}\), and the contraction minor \(\mathcal F/e\) is defined as the inclusion-wise minimal sets in \(\{S-\{e\}:S\in\mathcal F\}\). A minor of \(\mathcal F\) is a clutter obtained by a sequence of contractions and deletions.

A signed matroid is a pair \((M,\Sigma)\) where \(M\) is a binary matroid and \(\Sigma\subseteq EM\). A cycle of \(M\) is even (resp. odd) if \(|C\cap\Sigma|\) is even (resp. odd). Clearly, the set of inclusion-wise minimal odd cycles of \((M,\Sigma)\) is a clutter. Clutters that can be obtained by this construction are said to be binary.

A clutter \(\mathcal F\) is Eulerian if the parity of the cardinality of all inclusion-wise minimal covers is the same. At first glance this condition may appear non intuitive, so let us analyze what it says for a clutter \(\mathcal F\) of \(T\)-cuts in a graph \(G\) with terminals \(T\). The inclusion-wise minimal covers of \(\mathcal F\) are \(T\)-joins, and the parity of all \(T\)-joins is the same exactly when the symmetric difference of any two \(T\)-joins is a set with even cardinality. As the difference of two \(T\)-joins is an Eulerian subgraph, the condition translates to the requirement that \(G\) be bipartite.

The Cycling Conjecture

An Eulerian binary clutter packs if it does not contain any of the following clutters as a minor,

the Lines of the Fano,
the odd circuits of \(K_5\),
the complement of the cuts of \(K_5\), and
the \(T\)-joins of the Petersen graph where \(T\) is the set of all vertices.

For (1) the groundset of the clutter are the elements of the Fano matroids, and the members the lines. For (2) and (3) the groundset are the edges of the complete graph on \(5\) vertices. Note for (2) the members are the triangles and the pentagons and for (3) the members are triangles together with an independent edge and complete graphs on four vertices. For (4) the groundset are the edges of the Petersen graph and the member subgraphs where every vertex has odd degree. This conjecture is hard as we will see that it implies the Four Colour Theorem.

A special case of the conjecture,

A signed-graph is a pair \((G,\Sigma)\) where \(G\) is a graph and \(\Sigma\) is a subset of the edges. We say that \(B\subseteq E(G)\) is even (resp. odd) if \(|B\cap\Sigma|\) is even (resp. odd). We call a subset of the edges of \((G,\Sigma)\) where \(s,t\in V(G)\) an odd \(st\)-walk if it is either an odd \(st\)-path, or it is the union of an even \(st\)-path \(P\) and an odd circuit \(C\) where \(P\) and \(C\) share at most one vertex. We consider the empty set to be an (even) \(st\)-path for the case where \(s=t\). Thus when \(s=t\), odd \(st\)-walks are odd circuits. We proved that the Cycling conjecture holds for the clutter of odd \(st\)-walks, more precisely

Theorem 1

An Eulerian clutter of odd \(st\)-walks packs if it does not contain any of the following clutters as a minor,

the Lines of the Fano,
the odd circuits of \(K_5\).

Suppose \((G,\Sigma)\) is a signed-graph with vertices \(s,t\) and let \(\mathcal F\) be the clutter of odd \(st\)-walks of \((G,\Sigma)\). Then \(\mathcal F\) is Eulerian if either \(s=t\) and the degree of every vertex is even, or \(s\neq t\) and \(\deg(s)-|\Sigma|\) and the degree of every vertex in \(V(G)-\{s,t\}\) are even. The theorem states that in that case either: the maximum number of pairwise (edge) disjoint odd \(st\)-walks is equal to the minimum number of edges needed to intersect all odd \(st\)-walk, or \(\mathcal F\) must have as a minor the clutter of the lines of the Fano or the clutter of odd circuits of \(K_5\).

When \(s=t\), the condition that \({\mathcal F}\) is Eulerian says that all vertices of \(G\) have even degree. Moreover, in that case odd \(st\)-walks are odd circuits. Hence, Theorem 1 becomes,

Theorem 2

Let \(G\) be a graph that does not contain \(K_5\) as an odd minor and where every vertex has even degree. Then the minimum number of edges needed to intersect all odd circuits is equal to the maximum number of pairwise disjoint odd circuits.

A blocking vertex (resp. blocking pair) is a vertex (resp. pair of vertices) that intersects every odd circuit. Next we indicate a number of classes of signed-graphs for which the clutter of odd \(st\)-walks does not have as a minor the lines of the Fano or the clutter of odd circuits of \(K_5\).

Thus for each of the classes of signed-graphs (1)-(6), as long as the Eulerian condition holds, the minimum number of edges needed to intersect all odd \(st\)-walk is equal to the maximum number of disjoint odd \(st\)-walks.

Corollary

Let \((H,T)\) be a graft with \(|T|\leq 4\). Suppose that every vertex of \(H\) not in \(T\) has even degree and that all the vertices in \(T\) have degrees of the same parity. Then the maximum number of pairwise disjoint \(T\)-joins is equal to the minimum size of a \(T\)-cut.

Applying Theorem 1 to class (2) we can derive the two-commodity flow of Hu, Rothschild and Whinston, namely,

Corollary

Let \(H\) be a graph with vertices \(s_1,s_2,t_1,t_2\) where \(s_1\neq t_1\), \(s_2\neq t_2\), all of \(s_1,t_1,s_2,t_2\) have the same parity,and all the other vertices have even degree. Then the maximum number of pairwise disjoint paths that are between \(s_i\) and \(t_i\) for some \(i=1,2\), is equal to the minimum size of an edge subset whose deletion removes all \(s_1t_1\)- and \(s_2t_2\)-paths.

(\(\star\)) start from a plane graph with exactly two faces of odd length and distinct vertices \(s\) and \(t\) and identify \(s\) and \(t\).

Corollary

Let \(H\) be a graph as in (\(\star\)) and suppose that the length of the shortest odd circuit is \(k\). Then there exists cuts \(B_1,\ldots,B_k\) such that every edge \(e\) is in at least \(k-1\) of \(B_1,\ldots,B_k.\)

Suppose that \(H\) is as in (\(\star\)) and is loopless. Then by Corollary 6, there exists cuts \(\delta(U_1), \delta(U_2)\) such that every edge is in \(\delta(U_1)\cup\delta(U_2)\). It follows that for all \(i,j\in[4]\) where \(i\neq j\), \(U_i\cap U_j\) is a stable set. Hence, \(H\) is \(4\)-colourable.

The following conjecture would generalize the previous result as well as the 4-colour theorem,

Conjecture

Let \(H\) be a graph that does not contain \(K_5\) as an odd minor and suppose that the length of the shortest odd circuit is \(k\). Then there exists cuts \(B_1,\ldots,B_k\) such that every edge \(e\) is in at least \(k-1\) of \(B_1,\ldots,B_k\).

It can be shown that this conjecture is another special case of the Cycling Conjecture. It is open for planar graphs.

Bertrand Guenin - the set covering problem

1. Perfect and Ideal clutters

Examples of ideal clutters,

Binary clutters

Known cases of the Flowing Conjecture

The Weak Flowing Conjecture

2. The Cycling Conjecture

Examples of clutters that pack,

The conjecture,

A special case of the conjecture,