On cardinality constrained polymatroids

Dual Consistent Systems of Linear Inequalities and Cardinality Constrained Polytopes

“…Hence P c 1 ,c 2 f 1 ,f 2 = P and our example given above disproves a conjecture of Maurras, Spiegelberg, and Stephan [8,11] for the cardinality-constrained polymatroid intersection.…”

“…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. The latter implies that a conjecture of Maurras, Spiegelberg, and Stephan [8,11] about a linear representation of the cardinality-constrained intersection of polymatroids does not hold in general.…”

“…This result has been generalized by Maurras, Spiegelberg, and Stephan [8,11] to cardinality constrained polymatroids as follows. Given a polymatroid rank function f : 2 S → R and an increasing sequence (c 1 , .…”

“…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

“…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]).…”