@misc{Stephan2008, author = {Stephan, R{\"u}diger}, title = {On the cardinality constrained matroid polytope}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10614}, number = {08-08}, year = {2008}, abstract = {Edmonds showed that the so-called rank inequalities and the nonnegativity constraints provide a complete linear description of the matroid polytope. By essentially adding Gr{\"o}tschel's cardinality forcing inequalities, 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 cardinality forcing inequalities can be reduced to that for the rank inequalities. We also give necessary and sufficient conditions for a cardinality forcing inequality to be facet defining.}, language = {en} } @misc{Stephan2008, author = {Stephan, R{\"u}diger}, title = {Cardinality Constrained Combinatorial Optimization: Complexity and Polyhedra}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11026}, number = {08-48}, year = {2008}, abstract = {Given a combinatorial optimization problem and a subset \$N\$ of natural numbers, we obtain a cardinality constrained version of this problem by permitting only those feasible solutions whose cardinalities are elements of \$N\$. In this paper we briefly touch on questions that addresses common grounds and differences of the complexity of a combinatorial optimization problem and its cardinality constrained version. Afterwards we focus on polytopes associated with cardinality constrained combinatorial optimization problems. Given an integer programming formulation for a combinatorial optimization problem, by essentially adding Gr{\"o}tschel's cardinality forcing inequalities, we obtain an integer programming formulation for its cardinality restricted version. Since the cardinality forcing inequalities in their original form are mostly not facet defining for the associated polyhedra, we discuss possibilities to strengthen them.}, language = {en} }