Skip to main content
Log in

Improved initial vertex ordering for exact maximum clique search

Applied Intelligence Aims and scope Submit manuscript

Cite this article

An Erratum to this article was published on 18 November 2016


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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others





  1. Konc J, Janezic D (2010) ProBiS algorithm for detection of structurally similar protein binding sites by local structural alignment. Bioinformatics 26:1160–1168

    Article  Google Scholar 

  2. Eblen J, Phillips C, Rogers G, Langston M (2012) The maximum clique enumeration problem: algorithms, applications, and implementations. BMC Bioinforma 13:S5

    Article  Google Scholar 

  3. Butenko S, Chaovalitwongse W, Pardalos P (eds) (2009) Clustering challenges in biological networks. World Scientific, Singapore

  4. San Segundo P, Artieda J (2015) A novel clique formulation for the visual feature matching problem. Appl Intell 43(2):325–342

    Article  Google Scholar 

  5. San Segundo P, Rodriguez-Losada D (2013) Robust global feature based data association with a sparse bit optimized maximum clique algorithm. IEEE Trans Robot 29(5):1332–1339

    Article  Google Scholar 

  6. Östergård P (2002) A fast algorithm for the maximum clique problem. Discrete Appl Math 120:1:97–207

    Article  MathSciNet  Google Scholar 

  7. Fahle T (2002) Simple and fast: Improving a -and-bound algorithm for maximum clique. In: Proceedings ESA-2002, pp 485–498

  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 278–289

  9. Tomita E, Sutani Y, Higashi T, Takahashi S, Wakatsuki M (2010) A simple and faster branch-and-bound algorithm for finding a maximum clique. LNCS 5942:191–203

    MathSciNet  MATH  Google Scholar 

  10. San Segundo P, Rodriguez-Losada D, Jimenez A (2011) An exact bit-parallel algorithm for the maximum clique problem. Comput Oper Res 38:2:571–581

    Article  MathSciNet  MATH  Google Scholar 

  11. San Segundo P, Matia F, Rodriguez-Losada D, Hernando M (2013) An improved bit parallel exact maximum clique algorithm. Optim Lett 7:3:467–479

    Article  MathSciNet  MATH  Google Scholar 

  12. San Segundo P, Tapia C (2014) Relaxed approximate coloring in exact maximum clique search. Comput Oper Res 44:185–192

    Article  MathSciNet  MATH  Google 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 128–133

  14. Li C-M, Quan Z (2010) Combining Graph Structure Exploitation and Propositional Reasoning for the Maximum Clique Problem. In: Proceedings ICTAI, pp 344–351

  15. San Segundo P, Nikolaev A, Batsyn M (2015) Infra-chromatic bound for exact maximum clique search. Comput Oper Res 64:293–303

    Article  MathSciNet  Google Scholar 

  16. Konc J, Janečič D (2007) An improved branch and bound algorithm for the maximum clique problem. MATCH Commun Math Comput Chem 58:569–590

    MathSciNet  MATH  Google Scholar 

  17. Batsyn M, Goldengorin B, Maslov E, Pardalos P (2014) Improvements to MCS algorithm for the maximum clique problem. J Comb Optim 27:397–416

    Article  MathSciNet  MATH  Google 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 939–946

  19. Bron C, Kerbosch J (1973) Algorithm 457: finding all cliques of an undirected graph. Commun ACM 16:9:575–577

    Article  MATH  Google Scholar 

  20. Balas E, Yu C (1986) Finding a maximum clique in an arbitrary graph. SIAM J Comput 15:4:1054–1068

    Article  MathSciNet  MATH  Google Scholar 

  21. Prosser P (2012) Exact algorithms for maximum clique: a computational study. Algorithms 5:4:545–587

    Article  MathSciNet  Google Scholar 

  22. Carraghan R, Pardalos P (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9:6:375–382

    Article  MATH  Google Scholar 

  23. Personal communication with researchers Ciaran McCreesh and Patrick Prosser

  24. Welsh D, Powell M (1976) An upper bound for the chromatic number of a graph and its application to timetabling problem. Comput J 10:1:85–86

    MATH  Google Scholar 

  25. Syslo M (1989) Sequential coloring versus Welsh-Powell bound. Discret Math 74:241–243

    Article  MathSciNet  MATH  Google Scholar 

  26. Leighton F (1979) A graph coloring algorithm for large scheduling problems. J Res Natl Bur Stand 84 (6):489–506

    Article  MathSciNet  MATH  Google Scholar 

  27. Wu Q, Hao J (2015) A review on algorithms for maximum clique problems. Eur J Oper Res 242:3:693–709

    Article  MathSciNet  MATH  Google Scholar 

  28. Andrade D, Resende MG, Werneck R (2012) Fast local search for the maximum independent set problem. J Heuristics 18:4:525–547

    Article  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pablo San Segundo.

Additional information

An erratum to this article is available at



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.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Segundo, P.S., Lopez, A., Batsyn, M. et al. Improved initial vertex ordering for exact maximum clique search. Appl Intell 45, 868–880 (2016).

Download citation

  • Published:

  • Issue Date:

  • DOI: