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SMC-BRB: an algorithm for the maximum clique problem over large and sparse graphs with the upper bound via \(s^+\)-index

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Abstract

The Maximum Clique Problem (MCP) is a classic NP-hard problem, which has the goal of finding the largest possible clique. It is known to have direct applications in a wide spectrum of fields such as data association problems appearing in bioinformatics and computational biology, computer vision and robotics. Solutions like using brute force, backtracking and branch and bound are designed to deal with the maximum problem. In the branch and bound method, a branch is pruned if the currently found largest clique is better than its upper bound. However, the upper bound obtained by current methods is often not close enough to the \(\omega (G)\), leading to large inefficient search space. This paper discusses the branch and bound procedure to solve the maximum clique problem in large and sparse graphs and proposes a new efficient branch and bound maximum clique algorithm named SMC-BRB. SMC-BRB solves the maximum clique problem in heuristic search stage and exact search stage. It simultaneously utilizes the \(s^+\)-index and color-based upper bound in heuristic search stage, which effectively reduces the number of branches in the exact search stage. This method is beneficial to the solution of MCP because it provides a scale reduction on heuristic search stage. Experimental results show that SMC-BRB has better performance than the state-of-the-art algorithm MC-BRB, which demonstrates the efficiency of the proposed approach.

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Data availability

Datasets used in this paper are all available in public repositories.

Notes

  1. http://snap.stanford.edu/.

  2. http://networkrepository.com/.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Shimizu S, Yamaguchi K, Saitoh T, Masuda S (2017) Fast maximum weight clique extraction algorithm: optimal tables for branch-and-bound. Discret Appl Math 223:120–134

    Article  MathSciNet  MATH  Google Scholar 

  3. Blum C, Djukanovic M, Santini A, Jiang H, Raidl G (2021) Solving longest common subsequence problems via a transformation to the maximum clique problem. Comput Oper Res 125:105089

    Article  MathSciNet  MATH  Google Scholar 

  4. Belachew MT, Gillis N (2017) Solving the maximum clique problem with symmetric rank-one non-negative matrix approximation. J Optim Theory Appl 173(1):279–296

    Article  MathSciNet  MATH  Google Scholar 

  5. Chicco D (2017) Ten quick tips for machine learning in computational biology. BioData Mining 10(1):35

    Article  Google Scholar 

  6. Snead W, Hayden C, Gadok A, Rangamani P, Stachowiak J (2017) Membrane fission by protein crowding [Biophysics and Computational Biology]. Biophys J 112(3):327a

    Article  Google Scholar 

  7. Arshan Nasir, Mo Kim Kyung, Gustavo Caetano-Anollés (2018) Global patterns of protein domain gain and loss in superkingdoms. PLoS Comput Biol 10:1003452

    Google Scholar 

  8. Hao F, Min G, Pei Z, Park DS, Yang LT (2017) \(K\)-clique community detection in social networks based on formal concept analysis. IEEE Syst J 11:250

    Article  Google Scholar 

  9. Wang CC, Day MY, Lin YR (2016) Toward understanding the cliques of opinion spammers with social network analysis. In: 2016 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM)

  10. San Segundo P, Artieda J (2015) A novel clique formulation for the visual feature matching problem. Appl Intelli Int J Artif Intell Neural Netw Complex Problem-Solving Tech 43(2):325–342

    Google Scholar 

  11. Yang Y, Zhong Z, Shen T, Lin Z (2018) Convolutional neural networks with alternately updated clique. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)

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

    Google Scholar 

  13. Elmsallati A, Msalati A, Kalita J (2016) Index-based network aligner of protein-protein interaction networks. IEEE/ACM Trans Comput Biol Bioinform PP(99):330–336

    Google Scholar 

  14. Fn A, Fmp A, Bb B (2020) Detecting a most closeness-central clique in complex networks. Eur J Oper Res 283(2):461–475

    Article  MathSciNet  Google Scholar 

  15. Babel L, Tinhofer G (1990) A branch and bound algorithm for the maximum clique problem. Z Oper Res 34(3):207–217

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  17. Tomita Etsuji, Seki Tomokazu (2003) An efficient branch-and-bond algorithm for finding a maximum clique. Discret Math Theor Comput Sci. 278–289

  18. Tomita E, Kameda T (2007) An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J Global Optim 37(1):95–111

    Article  MathSciNet  MATH  Google Scholar 

  19. Brélaz D (1979) New methods to color the vertices of a graph. Commun ACM 22(4):251–256

    Article  MathSciNet  MATH  Google Scholar 

  20. Tomita E, Sutani Y, Higashi T, Wakatsuki M (2013) A simple and faster branch-and-bound algorithm for finding a maximum clique with computational experiments. IEICE Trans Inf Syst E96.D(6):1286–1298

    Article  MATH  Google Scholar 

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

  22. Li Chu-Min, Quan Zhe (2010) An efficient branch-and-bound algorithm based on MaxSAT for the maximum clique problem. In: Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence. 1:128–133

  23. Li Chu-Min, Fang Zhiwen, Xu Ke (2013) Combining MaxSAT reasoning and incremental upper bound for the maximum clique problem. In: 25th International Conference on Tools with Artificial Intelligence: 25th International Conference on Tools with Artificial Intelligence (ICTAI 2013), 4–6 November 2013, Washington, DC, USA, 939–946

  24. Li C-M, Fang Z, Jiang H, Xu K (2018) Incremental upper bound for the maximum clique problem. INFORMS J Comput 30(1):137–153

    Article  MathSciNet  MATH  Google Scholar 

  25. Calandriello D, Koutis I, Lazaric A, Valko M (2018) Improved large-scale graph learning through ridge spectral sparsification

  26. Wang L, Li CM, Zhou J, Jin B, Yin M (2019) An exact algorithm for minimum weight vertex cover problem in large graphs

  27. Shahinpour S, Shirvani S, Ertem Z, Butenko S (2017) Scale reduction techniques for computing maximum induced bicliques. Algorithms 10(4):113

    Article  MathSciNet  MATH  Google Scholar 

  28. Chang L (2020) Efficient maximum clique computation and enumeration over large sparse graphs. VLDB J Int J Very Large Data Bases 29(5):999–1022

    Article  Google Scholar 

  29. Rossi Ryan A, Gleich David F, Gebremedhin Assefaw H (2015) Parallel maximum clique algorithms with applications to network analysis. SIAM J Sci Comput 37(5):C589–C616

    Article  MathSciNet  MATH  Google Scholar 

  30. Pablo San Segundo, Alvaro Lopez, Pardalos Panos M (2016) A new exact maximum clique algorithm for large and massive sparse graphs. Comput Oper Res 66:81–94

    Article  MathSciNet  MATH  Google Scholar 

  31. Barber B, Kühn D, Lo A, Montgomery R, Osthus D (2017) Fractional clique decompositions of dense graphs and hypergraphs. J Combin Theory 127:148

    Article  MathSciNet  MATH  Google Scholar 

  32. Liang Z, Shan E, Kang L (2016) Clique-coloring claw-free graphs. Graphs Comb 32(4):1473–1488

    Article  MathSciNet  MATH  Google Scholar 

  33. Mohammadi N, Kadivar M (2020) NK-MaxClique and MMCQ: tow new exact branch and bound algorithms for the maximum clique problem. IEEE Access 8:180045–180053

    Article  Google Scholar 

  34. Segundo PS, Lopez A, Batsyn M, Nikolaev A, Pardalos PM (2016) Improved initial vertex ordering for exact maximum clique search. App Intell Int J Artif Intell Neural Netw Complex Problem Solv Technol 45:868

    Google Scholar 

  35. Kumlander D, Poroin A (2020) Reversed search maximum clique algorithm based on recoloring

  36. 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 

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Authors and Affiliations

  1. College of Computer Science, Chongqing University, Chongqing, 400044, China

    Mingqiang Zhou, Qianqian Zeng & Ping Guo

  2. College Key Laboratory of Software Theory and Technology, Chongqing, 400044, China

    Mingqiang Zhou & Ping Guo

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  1. Mingqiang Zhou
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  2. Qianqian Zeng
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Correspondence to Mingqiang Zhou.

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Zhou, M., Zeng, Q. & Guo, P. SMC-BRB: an algorithm for the maximum clique problem over large and sparse graphs with the upper bound via \(s^+\)-index. J Supercomput 79, 8026–8047 (2023). https://doi.org/10.1007/s11227-022-04982-7

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