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Semidefinite relaxations for partitioning, assignment and ordering problems

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  • F. Rendl

    (Alpen-Adria Universität Klagenfurt)

Abstract

Semidefinite optimization is a strong tool in the study of NP-hard combinatorial optimization problems. On the one hand, semidefinite optimization problems are in principle solvable in polynomial time (with fixed precision), on the other hand, their modeling power allows to naturally handle quadratic constraints. Contrary to linear optimization with the efficiency of the Simplex method, the algorithmic treatment of semidefinite problems is much more subtle and also practically quite expensive. This survey-type article is meant as an introduction for a non-expert to this exciting area. The basic concepts are explained on a mostly intuitive level, and pointers to advanced topics are given. We provide a variety of semidefinite optimization models on a selection of graph optimization problems and give a flavour of their practical impact.

Suggested Citation

  • F. Rendl, 2016. "Semidefinite relaxations for partitioning, assignment and ordering problems," Annals of Operations Research, Springer, vol. 240(1), pages 119-140, May.
    • Handle: RePEc:spr:annopr:v:240:y:2016:i:1:d:10.1007_s10479-015-2015-1
    DOI: 10.1007/s10479-015-2015-1
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    References listed on IDEAS

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    1. Rendl, F. & Sotirov, R., 2007. "Bounds for the quadratic assignment problem using the bundle method," Other publications TiSEM b6d298bc-77c9-4a6d-a043-5, Tilburg University, School of Economics and Management.
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    7. Frédéric Roupin, 2004. "From Linear to Semidefinite Programming: An Algorithm to Obtain Semidefinite Relaxations for Bivalent Quadratic Problems," Journal of Combinatorial Optimization, Springer, vol. 8(4), pages 469-493, December.
    8. Fischer, I. & Gruber, G. & Rendl, F. & Sotirov, R., 2006. "Computational experience with a bundle approach for semidenfinite cutting plane relaxations of max-cut and equipartition," Other publications TiSEM 03dfd8c3-9216-4c75-8921-3, Tilburg University, School of Economics and Management.
    9. Qing Zhao & Stefan E. Karisch & Franz Rendl & Henry Wolkowicz, 1998. "Semidefinite Programming Relaxations for the Quadratic Assignment Problem," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 71-109, March.
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    Cited by:

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    2. Kuryatnikova, Olga & Sotirov, Renata & Vera, J.C., 2022. "The maximum $k$-colorable subgraph problem and related problems," Other publications TiSEM 40e477c0-a78e-4ee1-92de-8, Tilburg University, School of Economics and Management.
    3. Zhi Pei & Mingzhong Wan & Ziteng Wang, 2020. "A new approximation algorithm for unrelated parallel machine scheduling with release dates," Annals of Operations Research, Springer, vol. 285(1), pages 397-425, February.

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