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New heuristics for the vertex coloring problem based on semidefinite programming


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.

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We thank to Igor Dukanovic for sharing his expertise and code for exact and approximate vertex coloring based on \(\varTheta \) function.

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Correspondence to Janez Povh.

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Supported by the Serbian Ministry of Education and Science (projects III 44006 and OI 174018), Supported by the Slovenian Research Agency (bilateral project no. BI-SR/10-11-040).

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Govorčin, J., Gvozdenović, N. & Povh, J. New heuristics for the vertex coloring problem based on semidefinite programming. Cent Eur J Oper Res 21, 13–25 (2013).

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  • Semidefinite programming
  • Vertex coloring problem
  • Boundary point method

Mathematics Subject Classification (2000)

  • 90C22
  • 13J30
  • 14P10
  • 47A57
  • 08B20