Combinatorial optimization: packing and covering

2002 ChoiceReviews  
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 [208] 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
more » ... 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 [130]) 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 [208] 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 [208] , 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 [41] 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 [133]), the weak Max Flow Min Cut property (Seymour [183]) or the Q + -MFMC property (Schrijver [172]). 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.
doi:10.5860/choice.39-4018 fatcat:rntd6v7wc5bwjneouupuhg4sr4