Stable Multi-Sets
Please always quote using this URN: urn:nbn:de:0297-zib-6047
- In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets. For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined. The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks.
Author: | Arie M.C.A. Koster, Adrian Zymolka |
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Document Type: | ZIB-Report |
Tag: | polyhedral combinatorics; stable multi-sets |
MSC-Classification: | 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C10 Integer programming |
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C27 Combinatorial optimization | |
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut | |
Date of first Publication: | 2000/11/03 |
Series (Serial Number): | ZIB-Report (00-36) |
ZIB-Reportnumber: | 00-36 |
Published in: | Appeared in: Mathematical Methods of Operations Research 56 (2002) 45 - 65 |