Improved initial vertex ordering for exact maximum clique search
- 259 Downloads
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).
KeywordsMaximum clique Branch-and-bound Approximate colouring Combinatorial optimization
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.
- 3.Butenko S, Chaovalitwongse W, Pardalos P (eds) (2009) Clustering challenges in biological networks. World Scientific, SingaporeGoogle Scholar
- 7.Fahle T (2002) Simple and fast: Improving a -and-bound algorithm for maximum clique. In: Proceedings ESA-2002, pp 485–498Google Scholar
- 8.Tomita E, Seki T (2003) An efficient branch and bound algorithm for finding a maximum clique. In: Calude C, Dinneen M, Vajnovszki V (eds) Discrete Mathematics and Theoretical Computer Science. LNCS, vol 2731, pp 278–289Google Scholar
- 13.Li C-M, Quan Z (2010) An Efficient Branch-and-Bound Algorithm based on MaxSAT for the Maximum Clique Problem. In: Proceedings AAAI, pp 128–133Google Scholar
- 14.Li C-M, Quan Z (2010) Combining Graph Structure Exploitation and Propositional Reasoning for the Maximum Clique Problem. In: Proceedings ICTAI, pp 344–351Google Scholar
- 18.Li C-M, Fang Z, Xu K (2013) Combining MaxSAT Reasoning and Incremental Upper Bound for the Maximum Clique Problem. In: Proceedings ICTAI, pp 939–946Google Scholar
- 23.Personal communication with researchers Ciaran McCreesh and Patrick ProsserGoogle Scholar