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Relaxations of Combinatorial Problems Via Association Schemes

In: Handbook on Semidefinite, Conic and Polynomial Optimization

Author

Listed:
  • Etienne Klerk

    (Tilburg University)

  • Fernando M. Oliveira Filho

    (Tilburg University)

  • Dmitrii V. Pasechnik

    (Nanyang Technological University)

Abstract

In this chapter we describe a novel way of deriving semidefinite programming relaxations of a wide class of combinatorial optimization problems. Many combinatorial optimization problems may be viewed as finding an induced subgraph of a specific type of maximum weight in a given weighted graph. The relaxations we describe are motivated by concepts from algebraic combinatorics. In particular, we consider a matrix algebra that contains the adjacency matrix of the required subgraph, and formulate a convex relaxation of this algebra. Depending on the type of subgraph, this algebra may be the Bose–Mesner algebra of an association scheme, or, more generally, a coherent algebra. Thus we obtain new (and known) relaxations of the traveling salesman problem, maximum equipartition problems in graphs, the maximum stable set problem, etc.

Suggested Citation

  • Etienne Klerk & Fernando M. Oliveira Filho & Dmitrii V. Pasechnik, 2012. "Relaxations of Combinatorial Problems Via Association Schemes," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 171-199, Springer.
    • Handle: RePEc:spr:isochp:978-1-4614-0769-0_7
    DOI: 10.1007/978-1-4614-0769-0_7
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    Citations

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    Cited by:

    1. Hu, Hao & Sotirov, Renata & Wolkowicz, Henry, 2023. "Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs," Other publications TiSEM 8dd3dbae-58fd-4238-b786-e, Tilburg University, School of Economics and Management.
    2. van Dam, E.R. & Sotirov, R., 2015. "On bounding the bandwidth of graphs with symmetry," Other publications TiSEM 180849f1-e7d3-44d9-8424-5, Tilburg University, School of Economics and Management.
    3. E. R. van Dam & R. Sotirov, 2015. "On Bounding the Bandwidth of Graphs with Symmetry," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 75-88, February.

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