The packing property. (English) Zbl 1033.90099

Cornuéjols, Gérard; Guenin, Bertrand; Margot, François

Math. Program. 89, No. 1 (A), 113-126 (2000).

Summary: A cluster \((V,E)\) packs if the smallest number of vertices needed to intersect all the edges (i.e. a minimum transversal) is equal to the maximum number of pairwise disjoint edges (i.e. a maximum matching). This terminology is due to P. D. Seymour [J. Comb. Theory, Ser. B 23, 189–222 (1977; Zbl 0375.05022)]. A clutter is minimally nonpacking if it does not pack but all its minors pack. An \(m\times n\) 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.

MSC:

90C27	Combinatorial optimization
05C70	Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)

Keywords:

max-flow min-cut property; cluster; minimally nonpacking