On the cardinality constrained matroid polytope

Abstract

Given a combinatorial optimization problemand an increasing finite sequence c of natural numbers, we obtain a cardinality constrained version c of by permitting only those feasible solutions of whose cardinalities are members of c. We are interested in polyhedra associated with those problems, in particular in inequalities that cut off solutions of forbidden cardinality. Maurras [9] and Camion and Maurras [1] introduced a family of inequalities, that we call forbidden set inequalities, which can be used to cut off those solutions. However, these inequalities are in general not facet defining for the polyhedron associated with c . In [7] it was shown how one can combine integer characterizations for cycle and path polytopes and a modified form of forbidden set inequalities to give facet defining integer representations for the cardinality restricted versions of these polytopes. Motivated by this work, we apply the same approach to the matroid polytope. It is well known that the so-called rank inequalities together with the nonnegativity constraints provide a complete linear description of the matroid polytope (see Edmonds [3]). By essentially adding the forbidden set inequalities in an appropriate form, we obtain a complete linear description of the cardinality constrained matroid polytope which is the convex hull of the incidence vectors of those independent sets that have a feasible cardinality. Moreover, we show how the separation problem for the forbidden set inequalities can be reduced to that for the rank inequalities. We also give necessary and sufficient conditions for a forbidden set inequality to be facet defining.

Dual consistent systems of linear inequalities and cardinality constrained polytopes

“…Cardinality constrained polyhedra and their linear representations were first investigated by Maurras [7] and Camion and Maurras [1], and later rediscovered by Grötschel [5] for what is called a cardinality homogeneous set system (also see related recent work by Kaibel and Stephan [6], Stephan [10], Maurras and Stephan [9], and Maurras, Spiegelberg, and Stephan [8]).…”

“…Recently Maurras and Stephan [9] derived strong valid inequalities that give a complete linear description for cardinality constrained matroids. This result has been generalized by Maurras, Spiegelberg, and Stephan [8,11] In the present paper we introduce the concept of dual consistent systems of linear inequalities and formulate the cardinality constrained problem in a more general setting.…”

“…Section 3 is concerned with multiple cardinality constraints. In Section 4 we show how the inequalities given in [9,8,11] are derived from our result. We also show that the systems of inequalities for the cardinality-constrained ordinary bipartite matching polytopes and for the cardinality-constrained (poly)matroid intersection are not dual consistent in general.…”

Dual Consistent Systems of Linear Inequalities and Cardinality Constrained Polytopes

“…Recently Maurras and Stephan [9] derived strong valid inequalities that give a complete linear description for cardinality constrained matroids. This result has been generalized by Maurras, Spiegelberg, and Stephan [8,11] to cardinality constrained polymatroids as follows.…”

“…Section 3 is concerned with multiple cardinality constraints. In Section 4 we show how the inequalities given in [9,8,11] are derived from our result. We also show that the systems of inequalities for the cardinality-constrained ordinary bipartite matching polytopes and for the cardinality-constrained (poly)matroid intersection are not dual consistent in general.…”