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b-Tree Facets for the Simple Graph Partitioning Polytope

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  • Michael M. Sørensen

    (The Aarhus School of Business)

Abstract

The simple graph partitioning problem is to partition an edge-weighted graph into mutually disjoint subgraphs, each consisting of no more than b nodes, such that the sum of the weights of all edges in the subgraphs is maximal. In this paper we introduce a large class of facet defining inequalities for the simple graph partitioning polytopes $$\mathcal{P}$$ n (b), b ≥ 3, associated with the complete graph on n nodes. These inequalities are induced by a graph configuration which is built upon trees of cardinality b. We provide a closed-form theorem that states all necessary and sufficient conditions for the facet defining property of the inequalities.

Suggested Citation

  • Michael M. Sørensen, 2004. "b-Tree Facets for the Simple Graph Partitioning Polytope," Journal of Combinatorial Optimization, Springer, vol. 8(2), pages 151-170, June.
    • Handle: RePEc:spr:jcomop:v:8:y:2004:i:2:d:10.1023_b:joco.0000031417.96218.26
    DOI: 10.1023/B:JOCO.0000031417.96218.26
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    References listed on IDEAS

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    1. Macambira, Elder Magalhaes & de Souza, Cid Carvalho, 2000. "The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations," European Journal of Operational Research, Elsevier, vol. 123(2), pages 346-371, June.
    2. Hunting, Marcel & Faigle, Ulrich & Kern, Walter, 2001. "A Lagrangian relaxation approach to the edge-weighted clique problem," European Journal of Operational Research, Elsevier, vol. 131(1), pages 119-131, May.
    3. FERREIRA, C. E. & MARTIN, A. & de SOUZA, C. C. & WEISMANTEL, R., 1996. "Formulations and valid inequalities for the node capacitated graph partitioning problem," LIDAM Reprints CORE 1236, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. FERREIRA, Carlos E. & MARTIN, Alexander & de SOUZA, Cid C. & WEISMANTEL, Robert, 1998. "The node capacitated graph partitioning problem: A computational study," LIDAM Reprints CORE 1335, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. M. Deza & M. Grötschel & M. Laurent, 1992. "Clique-Web Facets for Multicut Polytopes," Mathematics of Operations Research, INFORMS, vol. 17(4), pages 981-1000, November.
    6. Park, Kyungchul & Lee, Kyungsik & Park, Sungsoo, 1996. "An extended formulation approach to the edge-weighted maximal clique problem," European Journal of Operational Research, Elsevier, vol. 95(3), pages 671-682, December.
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