Combinatorial optimization: packing and covering
A clutter C is a pair (V, E), where V is a finite set and E is a family of subsets of V none of which is included in another. The elements of V are the vertices of C and those of E are the edges. For example, a simple graph (V, E) (no multiple edges or loops) is a clutter. We refer to West  for definitions in graph theory. In a clutter, a matching is a set of pairwise disjoint edges. A transversal is a set of vertices that intersects all the edges. A clutter is said to pack if the maximum
... ardinality of a matching equals the minimum cardinality of a transversal. This terminology is due to Seymour 1977. Many min-max theorems in graph theory can be rephrased by saying that a clutter packs. We give three examples. The first is König's theorem. König ) In a bipartite graph, the maximum cardinality of a matching equals the minimum cardinality of a transversal. As a second example, let s and t be distinct nodes of a graph G. Menger's theorem states that the maximum number of pairwise edgedisjoint st-paths in G equals the minimum number of edges in an st-cut (see West  Theorem 4.2.18). Let C 1 be the clutter whose vertices are the edges of G and whose edges are the st-paths of G (Following West's terminology  , paths and cycles have no repeated nodes). We call C 1 the clutter of st-paths. Its transversals are the st-cuts. Thus Menger's theorem states that the clutter of st-paths packs. Interestingly, some difficult results and famous conjectures can be rephrased by saying that certain clutters pack. As a third example, 1.1. MFMC PROPERTY AND IDEALNESS Definition 1.3 Clutter C(M ) packs if both (1.1) and (1.2) have optimal solution vectors x and y that are integral when w = 1. Definition 1.4 Clutter C(M ) has the packing property if both (1.1) and (1.2) have optimal solution vectors x and y that are integral for all vectors w with components equal to 0, 1 or +∞. Definition 1.5 Clutter C(M ) has the Max Flow Min Cut property (or MFMC property) if both (1.1) and (1.2) have optimal solution vectors x and y that are integral for all nonnegative integral vectors w. Clearly, the MFMC property for a clutter implies the packing property which itself implies that the clutter packs. Conforti and Cornuéjols  conjectured that, in fact, the MFMC property and the packing property are identical. This conjecture is still open. Conjecture 1.6 A clutter has the MFMC property if and only if it has the packing property. Definition 1.7 Clutter C(M ) is ideal if (1.1) has an optimal solution vector x that is integral for all w ≥ 0. The notion of idealness is also known as the width-length property (Lehman ), the weak Max Flow Min Cut property (Seymour ) or the Q + -MFMC property (Schrijver ). It is easy to show that the MFMC property implies idealness. Indeed, if (1.1) has an optimal solution vector x for all nonnegative integral vectors w, then (1.1) has an optimal solution x for all nonnegative rational vectors w and, since the rationals are dense in the reals, for all w ≥ 0. In fact, the packing property implies idealness. Theorem 1.8 If a clutter has the packing property, then it is ideal.