This paper describes a new initial vertexordering procedure NEW_SORT designed to enhance approximate-colour exact algorithms for the maximum clique problem (MCP). NEW_SORT considers two different vertex orderings: degree and colour-based. The degree-based vertex ordering describes an improvement over a well-known vertex ordering used by exact solvers. Moreover, colour-based vertex orderings for the MCP have been traditionally considered suboptimal with respect to degree-based ones. NEW_SORT chooses the “best” of the two orderings according to a new evaluation function. The reported experiments on graphs taken from public datasets show that a leading exact solver using NEW_SORT —and further enhanced with a strong initial solution— can improve its performance very significantly (sometimes even exponentially).
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Pablo San Segundo and Alvaro Lopez are funded by the Spanish Ministry of Economy and Competitiveness (grants ARABOT: DPI 2010-21247-C02-01 and NAVEGASE: DPI 2014-53525-C3-1-R). Mikhail Batsyn, Alexey Nikolaev, and Panos M. Pardalos are supported by the Laboratory of Algorithms and Technologies for Network Analysis, NRU HSE. We would also like to thank Jorge Artieda for his help with the experiments. Finally, we express our gratitude to Chu-Min Li for providing the source code of MaxCLQ.
An erratum to this article is available at http://dx.doi.org/10.1007/s10489-016-0862-3.
The list of instances from DIMACS and BHOSHLIB benchmarks employed in the reported results in Table 2 is:
C125.9, C250.9, Mann_a9, Mann_a27, Mann_a45, brock200_1/4, brock_400_1/4, c-fat200-1, c-fat200-2, c-fat200-5, c-fat500-1, c-fat500-2, c-fat500-5, c-fat500-10, dsjc500.1, dsjc500.5, dsjc1000.1, dsjc1000.5, frb30-15-1/5, gen200_p0.9_44, gen200_p0.9_55, hamming6-2, hamming6-4, hamming8-2, hamming8-4, hamming10-2, johnons8-2-4, johnons8-4-4, johnons16-2-4, keller4, p_hat300-1/3, p_hat500-1/3, p_hat300-1/3, p_hat700-1/3, p_hat1000-1/2, p_hat1500-1, san200_0.7_1/2, san200_0.9_1/3, san400_0.5_1, san400_0.7_1/3, san400_0.9_1, san1000, sanr200_0.7, sanr200_0.9, sanr400_0.5, sanr400_0.9.
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Segundo, P.S., Lopez, A., Batsyn, M. et al. Improved initial vertex ordering for exact maximum clique search. Appl Intell 45, 868–880 (2016). https://doi.org/10.1007/s10489-016-0796-9