Skip to main content
SpringerLink
Log in

An Enhanced Infra-Chromatic Bound for the Maximum Clique Problem

  • Conference paper
  • First Online:
Machine Learning, Optimization, and Big Data (MOD 2016)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 10122))

Included in the following conference series:

  • 2600 Accesses

Abstract

There has been a rising interest in experimental exact algorithms for the maximum clique problem because the gap between the expected theoretical performance and the reported results in practice is becoming surprisingly large. One reason for this is the family of bounding functions denoted as infra-chromatic because they produce bounds which can be lower than the chromatic number of the bounded subgraph. In this paper we describe a way to enhance exact solvers with an additional infra-chromatic bounding function and report performance over a number of graphs from well known data sets. Moreover, the reported results show that the new enhanced procedure significantly outperforms state-of-the-art.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    http://cs.hbg.psu.edu/txn131/clique.html.

  2. 2.

    http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/graph-benchmarks.htm.

References

  1. Bomze, I., Budinich, M., Pardalos, P., Pelillo, M.: The maximum clique problem. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, vol. 4, pp. 1–74. Springer, New York (1999)

    Chapter  Google Scholar 

  2. Butenko, S., Chaovalitwongse, W., Pardalos, P. (eds.): Clustering Challenges in Biological Networks. World Scientific, Singapore (2009)

    Google Scholar 

  3. San Segundo, P., Rodriguez-Losada, P., Matia, D., Galan, R.: Fast exact feature based data correspondence search with an efficient bit-parallel MCP solver. Appl. Intell. 32(3), 311–329 (2010)

    Article  Google Scholar 

  4. San Segundo, P., Rodriguez-Losada, D.: Robust global feature based data association with a sparse bit optimized maximum clique algorithm. IEEE Trans. Robot. 99, 1–7 (2013)

    Google Scholar 

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

    Article  Google Scholar 

  6. Fahle, T.: Simple and fast: improving a branch-and-bound algorithm for maximum clique. In: Möhring, R., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 485–498. Springer, Heidelberg (2002). doi:10.1007/3-540-45749-6_44

    Chapter  Google Scholar 

  7. Tomita, E., Seki, T.: An efficient branch-and-bound algorithm for finding a maximum clique. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds.) DMTCS 2003. LNCS, vol. 2731, pp. 278–289. Springer, Heidelberg (2003). doi:10.1007/3-540-45066-1_22

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Tomita, E., Sutani, Y., Higashi, T., Takahashi, S., Wakatsuki, M.: A simple and faster branch-and-bound algorithm for finding a maximum clique. In: Rahman, Md.S, Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 191–203. Springer, Heidelberg (2010). doi:10.1007/978-3-642-11440-3_18

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, C.-M., Quan, Z.: Combining graph structure exploitation and propositional reasoning for the maximum clique problem. In: Proceedings of ICTAI, pp. 344–351 (2010)

    Google Scholar 

  15. Li, C.-M., Fang, Z., Xu, K.: Combining MaxSAT reasoning and incremental upper bound for the maximum clique problem. In: Proceedings of ICTAI, pp. 939–946 (2013)

    Google Scholar 

  16. Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs and nonapproximability — towards tight results. In: 1995 Proceedings of 36th Annual Symposium on Foundations of Computer Science, pp. 422–431. IEEE (1995)

    Google Scholar 

  17. Moon, J., Moser, L.: On cliques in graphs. Isr. J. Math. 3(1), 23–28 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tarjan, R.E., Trojanowski, A.E.: Finding a maximum independent set. Technical report, Computer Science Department, School of Humanities and Sciences, Stanford University, Stanford, CA, USA (1976)

    Google Scholar 

  19. Robson, J.M.: Finding a maximum independent set in time \( O(2^{{{n \mathord{\left/ {\vphantom {n 4}} \right. \kern-0pt} 4}}} ) \). Technical report 1251-01, LaBRI, Université de Bordeaux I (2001)

    Google Scholar 

  20. Lavnikevich, N.: On the complexity of maximum clique algorithms: usage of coloring heuristics leads to the \( \Omega \left( {2^{{{n \mathord{\left/ {\vphantom {n 5}} \right. \kern-0pt} 5}}} } \right) \) algorithm running time lower bound (2013)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work is funded by the Spanish Ministry of Economy and Competitiveness (grant NAVEGASE: DPI 2014-53525-C3-1-R).

Author information

Authors and Affiliations

  1. Center for Automation and Robotics (CAR), Polytechnic University of Madrid (UPM), C/ Jose Gutiérrez Abascal, 2, 28006, Madrid, Spain

    Pablo San Segundo, Jorge Artieda, Rafael Leon & Cristobal Tapia

Authors
  1. Pablo San Segundo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Artieda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rafael Leon
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cristobal Tapia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pablo San Segundo .

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Semantic Technology Laboratory, National Research Council (CNR), Catania, Italy

    Piero Conca

  3. Dipartimento di Sociologia e Metodi della Ricerca Sociale, Università di Catania, Catania, Italy

    Giovanni Giuffrida

  4. Department of Mathematics and Computer Science, University of Catania, Catania, Italy

    Giuseppe Nicosia

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

San Segundo, P., Artieda, J., Leon, R., Tapia, C. (2016). An Enhanced Infra-Chromatic Bound for the Maximum Clique Problem. In: Pardalos, P., Conca, P., Giuffrida, G., Nicosia, G. (eds) Machine Learning, Optimization, and Big Data. MOD 2016. Lecture Notes in Computer Science(), vol 10122. Springer, Cham. https://doi.org/10.1007/978-3-319-51469-7_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-51469-7_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-51468-0

  • Online ISBN: 978-3-319-51469-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation