The Packing Property

A clutter (V;E) packs if the smallest number of vertices needed to intersect all the edges (i.e. a transversal) is equal to the maximum number of pairwise disjoint edges (i.e. a matching). This terminology is due to Seymour 1977. A clutter is minimally nonpacking if it does not pack but all its minors pack. A 0,1 matrix is minimally nonpacking if it is the edge-vertex incidence matrix of a minimally nonpacking clutter. Minimally nonpacking matrices can be viewed as the counterpart for the set covering problem of minimally imperfect matrices for the set packing problem. This paper proves several properties of minimally nonpacking clutters and matrices. A clutter C is a pair V (C); E(C) , where V (C) is a finite set and E(C) = fS1; : : : ;Smg is a family of subsets of V (C) with the property that Si Sj implies Si = Sj . The elements of V (C) are the vertices of C and those of E(C) are the edges. A transversal of C is a minimal subset of vertices that intersects all the edges. Let (C) denote the cardinality of a smallest transversal. A clutter C packs if there exist (C) pairwise disjoint edges. For j 2 V (C), the contraction C=j and deletion C n j are clutters defined as follows: both have V (C) fjg as vertex set, E(C=j) is the set of minimal elements of fS fjg : S 2 E(C)g and E(C n j) = fS : j 62 S 2 E(C)g. Contractions and deletions of distinct vertices can be performed sequentially, and it is well known that the result does not depend on the order. A clutter D obtained from C by deleting Id V (C) and contracting Ic V (C), where Ic \ Id = ; and Ic [ Id 6= ;, is a minor of C and is denoted by C n Id=Ic. We say that a clutter C has the packing property if it packs and all its minors pack. A clutter is minimally non packing (mnp) if it does not pack but all its minors do. In this paper, we study mnp clutters. Date: November, 1997.