Cardinality constrained combinatorial optimization: Complexity and polyhedra

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 Grötschel’s cardinality forcing inequalities [11], 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. In [13] a variation of the cardinality forcing inequalities were successfully integrated in the system of linear inequalities for the matroid polytope to provide a complete linear description of the cardinality constrained matroid polytope. We identify this polytope as a master polytope for our class of problems, since many combinatorial optimization problems can be formulated over the intersection of matroids.