Fast Solving Maximum Weight Clique Problem in Massive Graphs

This paper explores techniques for fast solving the maximum weight clique problem (MWCP) in very large scale real-world graphs. Because of the size of such graphs and the intractability of MWCP, previously developed algorithms may not be applicable. Although recent heuristic algorithms make progress in solving MWCP in massive graphs, they still need considerable time to get a good solution. In this work, we propose a new method for MWCP which interleaves between clique construction and graph reduction. We also propose three novel ideas to make it efficient, and develop an algorithm called FastWClq. Experiments on massive graphs from various applications show that, Fast-WClq finds better solutions than state of the art algorithms while the run time is much less. Further, FastWClq proves the optimal solution for about half of the graphs in an averaged time less than one second.

[1]  Alok Singh,et al.  A Hybrid Evolutionary Approach to Maximum Weight Clique Problem , 2006 .

[2]  M. Haack.  Part B , 1942 .

[3]  Ryan A. Rossi,et al.  The Network Data Repository with Interactive Graph Analytics and Visualization , 2015, AAAI.

[4]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[5]  Isaac Siwale.  ON GLOBAL OPTIMIZATION , 2015 .

[6]  Wayne J. Pullan,et al.  Approximating the maximum vertex/edge weighted clique using local search , 2008, J. Heuristics.

[7]  Etsuji Tomita,et al.  An Efficient Branch-and-bound Algorithm for Finding a Maximum Clique with Computational Experiments , 2001, J. Glob. Optim..

[8]  Hua Jiang,et al.  Combining MaxSAT Reasoning and Incremental Upper Bound for the Maximum Clique Problem , 2013, 2013 IEEE 25th International Conference on Tools with Artificial Intelligence.

[9]  Shaowei Cai,et al.  Balance between Complexity and Quality: Local Search for Minimum Vertex Cover in Massive Graphs , 2015, IJCAI.

[10]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[11]  PullanWayne.  Approximating the maximum vertex/edge weighted clique using local search , 2008 .

[12]  Shinya Takahashi,et al.  A Simple and Faster Branch-and-Bound Algorithm for Finding a Maximum Clique , 2010, WALCOM.

[13]  Azriel Rosenfeld,et al.  Computer Vision , 1988, Adv. Comput..

[14]  Jin-Kao Hao,et al.  Breakout Local Search for maximum clique problems , 2013, Comput. Oper. Res..

[15]  Janez Konc,et al.  An improved branch and bound algorithm for the maximum clique problem , 2007 .

[16]  Wayne J. Pullan,et al.  Phased local search for the maximum clique problem , 2006, J. Comb. Optim..

[17]  Alok Singh,et al.  A hybrid heuristic for the maximum clique problem , 2006, J. Heuristics.

[18]  Ryan A. Rossi,et al.  Coloring large complex networks , 2014, Social Network Analysis and Mining.

[19]  Marcello Pelillo,et al.  A Complementary Pivoting Approach to the Maximum Weight Clique Problem , 2002, SIAM J. Optim..

[20]  Ram Dantu,et al.  An Impatient Evolutionary Algorithm With Probabilistic Tabu Search for Unified Solution of Some NP-Hard Problems in Graph and Set Theory via Clique Finding , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[21]  Anurag Verma,et al.  Solving the Maximum Clique and Vertex Coloring Problems on Very Large Sparse Networks , 2015, INFORMS J. Comput..

[22]  Nicholas I. M. Gould,et al.  SIAM Journal on Optimization , 2012 .

[23]  Etsuji Tomita,et al.  An Efficient Branch-and-Bound Algorithm for Finding a Maximum Clique , 2003, DMTCS.

[24]  Pablo Moscato,et al.  Identification of a 5-Protein Biomarker Molecular Signature for Predicting Alzheimer's Disease , 2008, PloS one.

[25]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .

[26]  Abdul Sattar,et al.  Local search with edge weighting and configuration checking heuristics for minimum vertex cover , 2011, Artif. Intell..

[27]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[28]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[29]  Pablo San Segundo,et al.  An improved bit parallel exact maximum clique algorithm , 2013, Optim. Lett..

[30]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[31]  Wayne J. Pullan,et al.  Dynamic Local Search for the Maximum Clique Problem , 2011, J. Artif. Intell. Res..

[32]  Aravind Srinivasan,et al.  Structural and algorithmic aspects of massive social networks , 2004, SODA '04.

[33]  Minghao Yin,et al.  Two Efficient Local Search Algorithms for Maximum Weight Clique Problem , 2016, AAAI.

[34]  Cristian S. Calude,et al.  Discrete Mathematics and Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[35]  Fred W. Glover,et al.  Multi-neighborhood tabu search for the maximum weight clique problem , 2012, Annals of Operations Research.

[36]  Chu Min Li,et al.  An efficient branch-and-bound algorithm based on MaxSAT for the maximum clique problem , 2010, AAAI 2010.

[37]  ScienceDirect.  Electronic notes in discrete mathematics , 1999 .

[38]  Stanislav Busygin,et al.  A new trust region technique for the maximum weight clique problem , 2006, Discret. Appl. Math..

[39]  Gilbert Laporte,et al.  Annals of Operations Research , 1996 .

[40]  Patric R. J. Östergård,et al.  A New Algorithm for the Maximum-Weight Clique Problem , 1999, Electron. Notes Discret. Math..

[41]  D. M. Deighton,et al.  Computers in Operations Research , 1977, Aust. Comput. J..

[42]  Vladimir Batagelj,et al.  An O(m) Algorithm for Cores Decomposition of Networks , 2003, ArXiv.

[43]  Sergiy Butenko,et al.  Graph Domination, Coloring and Cliques in Telecommunications , 2006, Handbook of Optimization in Telecommunications.

[44]  Ryan A. Rossi,et al.  Fast maximum clique algorithms for large graphs , 2014, WWW.

[45]  Ke Xu,et al.  Solving Maximum Weight Clique Using Maximum Satisfiability Reasoning , 2014, ECAI.

[46]  Stephen B. Seidman,et al.  Network structure and minimum degree , 1983 .

[47]  Peter Willett,et al.  MATCH Communications in Mathematical and in Computer Chemistry , 2016 .

[48]  L. Babai,et al.  Theory of Computing , 2015 .