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New heuristics for the vertex coloring problem based on semidefinite programming

Author

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  • Jelena Govorčin
  • Nebojša Gvozdenović
  • Janez Povh

Abstract

The Lovász theta number Lovász (IEEE Trans Inf Theory 25:1–7, 1979 ) is a well-known lower bound on the chromatic number of a graph $$G$$ , and $$\varPsi _K(G)$$ is its impressive strengthening Gvozdenović and Laurent (SIAM J Optim 19(2):592–615, 2008 ). The bound $$\varPsi _K(G)$$ was introduced in very specific and abstract setting which is tough to translate into usual mathematical programming framework. In the first part of this paper we unify the motivations and approaches to both bounds and rewrite them in a very similar settings which are easy to understand and straightforward to implement. In the second part of the paper we provide explanations how to solve efficiently the resulting semidefinite programs and how to use optimal solutions to get good coloring heuristics. We propose two vertex coloring heuristics based on $$\varPsi _K(G)$$ and present numerical results on medium sized graphs. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Jelena Govorčin & Nebojša Gvozdenović & Janez Povh, 2013. "New heuristics for the vertex coloring problem based on semidefinite programming," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(1), pages 13-25, June.
    • Handle: RePEc:spr:cejnor:v:21:y:2013:i:1:p:13-25
    DOI: 10.1007/s10100-012-0276-1
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    References listed on IDEAS

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    1. Gvozdenovic, N. & Laurent, M. & Vallentin, F., 2009. "Block-diagonal semidenfinite programming hierarchies for 0/1 programming," Other publications TiSEM 32708817-eaaf-4337-9962-b, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Andrej Kastrin & Janez Povh & Lidija Zadnik Stirn & Janez Žerovnik, 2021. "Methodologies and applications for resilient global development from the aspect of SDI-SOR special issues of CJOR," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 29(3), pages 773-790, September.
    2. Tibor Csendes & Lidija Zadnik Stirn & Janez Žerovnik, 2013. "Editorial," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(1), pages 1-2, June.
    3. Zehui Shao & Aleksander Vesel, 2015. "$$L(3,2,1)$$ L ( 3 , 2 , 1 ) -labeling of triangular and toroidal grids," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 23(3), pages 659-673, September.

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