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Cardinality Constrained Combinatorial Optimization: Complexity and Polyhedra

Please always quote using this URN: urn:nbn:de:0297-zib-11026
  • 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, 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.

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Metadaten
Author:Rüdiger Stephan
Document Type:ZIB-Report
Tag:Cardinality Constraints; Cardinality Forcing Inequalities; Combinatorial Optimization; Matroids
MSC-Classification:05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Bxx Designs and configurations (For applications of design theory, see 94C30) / 05B35 Matroids, geometric lattices [See also 52B40, 90C27]
05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C38 Paths and cycles [See also 90B10]
52-XX CONVEX AND DISCRETE GEOMETRY / 52Bxx Polytopes and polyhedra / 52B40 Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) [See also 05B35, 52Cxx]
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C27 Combinatorial optimization
Date of first Publication:2008/12/10
Series (Serial Number):ZIB-Report (08-48)
ISSN:1438-0064
ZIB-Reportnumber:08-48
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