A local core number based algorithm for the maximum clique problem

Document Type : Research Paper


Department of computer science, University of Shahrekord, Shahrekord, Iran


‎The maximum clique problem (MCP) is to determine a complete subgraph of maximum cardinality in a graph‎. ‎MCP is a fundamental problem in combinatorial optimization and is noticeable for its wide range of applications‎. ‎In this paper‎, ‎we present two branch-and-bound exact algorithms for finding a maximum clique in an undirected graph‎. ‎Many efficient exact branch and bound maximum clique algorithms use approximate coloring to compute an upper bound on the clique number but‎, ‎as a new pruning strategy‎, ‎we show that local core number is more efficient‎. ‎Moreover‎, ‎instead of neighbors set of a vertex‎, ‎our search area is restricted to a subset of the set in each subproblem which speeds up clique finding process‎. ‎This subset is based on the core of the vertices of a given graph‎. ‎We improved the MCQ and MaxCliqueDyn algorithms with respect to the new pruning strategy and search area restriction‎. ‎Experimental results demonstrate that the improved algorithms outperform the previous well-known algorithms for many instances when applied to DIMACS benchmark and random graphs‎.


Main Subjects

[1] P. San Segundo and J. Artieda, A novel clique formulation for the visual feature matching problem, Appl. Intell., 43
(2015) 325–342.
[2] P. San Segundo, D. Rodriguez-Losada, F. Matia and R. Galan, Fast exact feature based data correspondence search
with an efficient bit-parallel MCP solver, Appl. Intell., 32 (2010) 311–329.
[3] S. Butenko and W. E. Wilhelm, Clique-detection models in computational biochemistry and genomics, European J.
Operational Researc, 173 (2006) 1–17.
[4] C. W. Art, B. Sergiy and P. Panos M., Clustering challenges in biological networks, World Scientific Publishing Co,
Inc., 2009.
[5] T. Etzion and P. R. Ostergard, Greedy and heuristic algorithms for codes and colorings, IEEE Trans. Inform. Theor,
44 (1998) 382–388.
[6] T. Gschwind, S. Irnich, F. Furini and R. W. Calvo, Social network analysis and community detection by decomposing
a graph into relaxed cliques, Technical Report LM-2015-07; Chair of Logistics Management, Gutenberg School of
Management and Economics, Johannes Gutenberg University Mainz: Mainz, Germany, 2015.
[7] O. Weide, D. Ryan and M. Ehrgott, An iterative approach to to robust and integrated aircraft routing and crew
scheduling, Comput. Oper. Res., 37 (2010) 833–844.
[8] Q. Wu and J. K. Hao, A review on algorithms for maximum clique problems, European J. Oper. Res.h, 242 (2015)
[9] V. Batagelj and M. Zaversnik, O(m) algorithm for cores decomposition of networks, arXiv preprint cs/0310049, 2003
[10] S. B. Seidman, Network structure and minimum degree, Social Networks, 5 (1983) 269–287.
[11] M. R. Garey and D. S. Johnson, Computers and intractability: A guide to the theory of NP-completeness, A Series of
Books in the Mathematical Sciences. W. H. Freeman and Co., San Francisco, Calif., 1979.
[12] J. Hastad, Clique is hard to approximate within n1−ϵ , Acta Math., 182 (1999) 105–142.
[13] U. Feige, Approximating maximum clique by removing subgraphs, SIAM J. Discrete Math., 18 (2004) 219–225.
[14] P. R. J. Ostergard, A fast algorithm for the maximum clique problem, Discrete Appl. Math., 120 (2002) 197–207.
[15] L. Babel and G. Tinhofer, A branch and bound algorithm for the maximum clique problem, Z. Oper. Res., 34 (1990)
[16] D. Brelaz, New methods to color the vertices of a graph, Comm. ACM, 22 (1979) 251–256.
[17] E. Tomita and T. Seki, An efficient branch-and-bound algorithm for finding a maximum clique. In:Proceedings of
the discrete mathematics and theoretical computer science. Lecture Notes in Computer Science, 2731 (2003) 278–289.
[18] E. Tomita and T. Kameda, An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments, J. Global Optim., 37 (2007) 95–111.
[19] J. Konc and D. Janezic, An improved branch and bound algorithm for the maximum clique problem, MATCH
Commun. Math. Comput. Chem., 58 (2007) 569–590.
[20] P. San Segundo, D. Rodrguez-Losada and A. Jimenez, An exact bit-parallel algorithm for the maximum clique
problem, Comput. Oper. Res., 38 (2011) 571–581.
[21] E. Tomita, Y. Sutani, T. Higashi, S. Takahashi and M. Wakatsuki, A simple and faster branch-and bound algorithm
for finding a maximum clique, Lecture Notes in Comput. Sci., 5942 (2010) 191–203.
[22] P. San Segundo, F. Matia, D. Rodriguez-Losada and M. Hernando, An improved bit parallel exact maximum clique
algorithm, Optim. Lett., 7 (2013) 467–479.
[23] P. San Segundo, C. Tapia, Relaxed approximate coloring in exact maximum clique search, Comput. Oper. Res., 44
(2014) 185–192.
[24] R. A. Rossi, D. F. Gleich, A. H. Gebremedhin and M. M. A. Patwary, Fast maximum clique algorithms for large
graphs, In: Proceedings of the 23rd International Conference on World wide web, ACM, (2014) 365–366.
[25] R. A. Rossi, D. F. Gleich and A. H. Gebremedhin, Parallel maximum clique algorithms with applications to network
analysis, SIAM J. Sci. Comput., 37 (2015) C589–C616.
[26] B. Pattabiraman, M. M. A. Patwary, A. H. Gebremedhin, W. K. Liao and A. Choudhary, Fast algorithms for the
maximum clique problem on massive sparse graphs, Algorithms and models for the web graph, Lecture Notes in
Comput. Sci., 8305, Springer, Cham, (2013) 156-169
[27] P. San Segundo, A. Lopez and P. M. Pardalos, A new exact maximum clique algorithm for large and massive sparse
graphs, Comput. Oper. Res., 66 (2016) 81–94.
[28] P. San Segundo, A. Nikolaev and M. Batsyn, Infra-chromatic bound for exact maximum clique search, Comput. Oper.
Res., 64 (2015) 293–303.
[29] C. M. Li and Z. Quan, An efficient branch-and-bound algorithm based on MaxSAT for the maximum clique problem,
In: Proceedings of the 24th AAAI Conference on Artificial Intelligence, AAAI-10, (2010) 128–133.
[30] D. J. Welsh and M. B. Powell, An upper bound for the chromatic number of a graph and its application to timetabling problems, Comput. J., 10 (1967) 85–86.
[31] J. T. Hungerford and F. Rinaldi, A General Regularized Continuous Formulation for the Maximum Clique Problem,
Math. Oper. Res., 44 (2019) 1161–1173.
[32] K. Kanazawa and S. Cai, FPGA Acceleration to Solve Maximum Clique Problems Encoded into Partial MaxSAT,
2018 IEEE 12th International Symposium on Embedded Multicore/Many-core Systems-on-Chip (MCSoC), (2018)
[33] M. T. Belachew and N. Gillis, Solving the Maximum Clique Problem with Symmetric Rank-One Non-negative Matrix
Approximation, J. Optim. Theory Appl., 173 (2017) 279–296.
[34] A. Verma, A. Buchanan and S. Butenko, Solving the maximum clique and vertex coloring problems on very large
sparse networks, INFORMS J. Comput., 27 (2015) 164–177.
[35] C. M. Li, Z. Fang, H. Jiang and K. Xu, Incremental upper bound for the maximum clique problem, INFORMS J.
Comput., 30 (2017) 137–153.
Volume 10, Issue 3 - Serial Number 3
September 2021
Pages 149-163
  • Receive Date: 24 November 2019
  • Revise Date: 30 September 2020
  • Accept Date: 21 January 2021
  • Published Online: 01 September 2021