An efficient local search algorithm for solving maximum edge weight clique problem in large graphs

Yi Chu
Yi Chu
Boxiao Liu
Boxiao Liu
Haihang You
Haihang You

Journal of Combinatorial Optimization, pp. 933-954, 2020.

Cited by: 1|Bibtex|Views47|Links
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Keywords:
Graph reduction Maximum edge weight clique problem Stochastic local search
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We present an effective local search algorithm named ReConSLS, which works in three phases, i.e. clique construction, stochastic local search and graph reduction

Abstract:

Maximum vertex weight clique problem (MVWCP) and maximum edge weight clique problem (MEWCP) are two significant generalizations of maximum clique problem (MCP), and can be widely used in many real-world applications including molecular biology, broadband network design and pattern recognition. Recently, breakthroughs have been made for so...More

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Introduction
  • Due to the theoretical significance and practical application value, considerable efforts have been made to develop algorithms for MCP, MVWCP and MEWCP.
  • Algorithms for solving these problems are usually categorized into two classes: complete algorithms and incomplete algorithms
Highlights
  • Given an undirected graph G = (V, E), where V is the set of vertices and E is the set of edges, a clique C is a subset of V, such that all vertices in C are connected
  • We propose a new algorithm for maximum edge weight clique problem (MEWCP), which works in three phases, i.e., clique construction, local search and graph reduction
  • To evaluate the effectiveness of the graph reduction algorithm proposed in Sect. 3.2, as well as the techniques utilized in the clique construction phase and the stochastic local search phase, we disable the graph reduction algorithm in ReConSLS, resulting in another solver named ConSLS
  • We present an effective local search algorithm named ReConSLS, which works in three phases, i.e. clique construction, stochastic local search and graph reduction
  • We are devoted to improving the performance over the state-of-the-art algorithms for solving MEWCP on large graphs
  • We conducted experiments to compare ConSLS against ReConSLS, CERS, LSMR, LSCC and LSCC + Best from Multiple Selections’ (BMS), and the related results show that ConSLS is the second best solver following ReConSLS, indicating the effectiveness of our upper bound function and the techniques utilized in the clique construction phase and the stochastic local search phase
Methods
  • The authors include four state-of-the-art SLS solvers as the competitors. CERS (Fan et al 2017b) and LSMR (Li et al 2018) are two efficient solvers for MEWCP in large graphs.
  • Breakthroughs have been made in MVWCP solving, resulting in several state-of-the-art solvers, such as LSCC, LSCC + BMS (Wang et al 2016), FastWClq (Cai and Lin 2016) and WLMC (Jiang et al 2017).
  • The authors implemented ReConSLS, LSCC and LSCC + BMS in C++.
  • CERS was implemented in C++ by its author and could be downloaded online5.
  • All the experiments in this paper were carried out on a workstation under the operating system CentOS, with Intel(R) Xeon(R) CPU E5-2620 2.10GHz CPU, 20MB L3 cache and 128GB RAM
Results
  • Table 1 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on graphs from Network Repository.
  • Table 3 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on 18 graphs from KONECT.
  • ReConSLS finds the best clique weight on all 10 runs for all 18 graphs, while this figure for CERS, LSMR, LSCC and LSCC + BMS is only 13, 13, 14 and 13, respectively.
  • On 8 of the 18 graphs, ReConSLS’s speed of finding the best averaged clique weight is more than 3 times as fast as the second fastest competitor’s
Conclusion
  • The authors present an effective local search algorithm named ReConSLS, which works in three phases, i.e. clique construction, stochastic local search and graph reduction.
  • The authors conducted experiments to compare ConSLS against ReConSLS, CERS, LSMR, LSCC and LSCC + BMS, and the related results show that ConSLS is the second best solver following ReConSLS, indicating the effectiveness of the upper bound function and the techniques utilized in the clique construction phase and the stochastic local search phase.
Summary
  • Introduction:

    Due to the theoretical significance and practical application value, considerable efforts have been made to develop algorithms for MCP, MVWCP and MEWCP.
  • Algorithms for solving these problems are usually categorized into two classes: complete algorithms and incomplete algorithms
  • Methods:

    The authors include four state-of-the-art SLS solvers as the competitors. CERS (Fan et al 2017b) and LSMR (Li et al 2018) are two efficient solvers for MEWCP in large graphs.
  • Breakthroughs have been made in MVWCP solving, resulting in several state-of-the-art solvers, such as LSCC, LSCC + BMS (Wang et al 2016), FastWClq (Cai and Lin 2016) and WLMC (Jiang et al 2017).
  • The authors implemented ReConSLS, LSCC and LSCC + BMS in C++.
  • CERS was implemented in C++ by its author and could be downloaded online5.
  • All the experiments in this paper were carried out on a workstation under the operating system CentOS, with Intel(R) Xeon(R) CPU E5-2620 2.10GHz CPU, 20MB L3 cache and 128GB RAM
  • Results:

    Table 1 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on graphs from Network Repository.
  • Table 3 presents the comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on 18 graphs from KONECT.
  • ReConSLS finds the best clique weight on all 10 runs for all 18 graphs, while this figure for CERS, LSMR, LSCC and LSCC + BMS is only 13, 13, 14 and 13, respectively.
  • On 8 of the 18 graphs, ReConSLS’s speed of finding the best averaged clique weight is more than 3 times as fast as the second fastest competitor’s
  • Conclusion:

    The authors present an effective local search algorithm named ReConSLS, which works in three phases, i.e. clique construction, stochastic local search and graph reduction.
  • The authors conducted experiments to compare ConSLS against ReConSLS, CERS, LSMR, LSCC and LSCC + BMS, and the related results show that ConSLS is the second best solver following ReConSLS, indicating the effectiveness of the upper bound function and the techniques utilized in the clique construction phase and the stochastic local search phase.
Tables
  • Table1: Experimental results on graphs from network repository (For the sake of space, we do not report on graphs that all the solvers find the same best clique weight within 25 s, we do not report ‘Wavg’ which is equal to ‘Wmax’)
  • Table2: Experimental results on 139 graphs from network repository
  • Table3: Comparative results of ReConSLS, CERS, LSMR, LSCC and LSCC + BMS on real world graphs from KONECT
  • Table4: Experimental results of ReConSLS, ConSLS, CERS, LSMR, LSCC and LSCC + BMS on all the graphs
Download tables as Excel
Funding
  • This work is partially supported by the National Key Research and Development Program of China under Grant 2017YFB0202502
  • Shaowei Cai is supported by Youth Innovation Promotion Association, Chinese Academy of Sciences (No 2017150)
Reference
  • Abramé A, Habet D, Toumi D (2017) Improving configuration checking for satisfiable random k-sat instances. Ann Math Artif Intell 79(1–3):5–24
    Google ScholarLocate open access versionFindings
  • Alidaee B, Glover F, Kochenberger G, Wang H (2007) Solving the maximum edge weight clique problem via unconstrained quadratic programming. Eur J Oper Res 181(2):592–597
    Google ScholarLocate open access versionFindings
  • Balasundaram B, Butenko S (2006) Graph domination, coloring and cliques in telecommunications. In: Resende MGC, Pardalos PM (eds) Handbook of optimization in telecommunications. Springer, Boston
    Google ScholarLocate open access versionFindings
  • Ballard DH, Brown CM (1982) Computer vision. Prenice-Hall, Englewood Cliffs Battiti R, Protasi M (2001) Reactive local search for the maximum clique problem 1. Algorithmica
    Google ScholarLocate open access versionFindings
  • 29(4):610–637 Benlic U, Hao JK (2013) Breakout local search for maximum clique problems. Comput Oper Res 40(1):192–
    Google ScholarLocate open access versionFindings
  • 206 Cai S (2015) Balance between complexity and quality: local search for minimum vertex cover in massive graphs. In: Proceedings of IJCAI 2015, pp 747–753 Cai S, Su K (2012) Configuration checking with aspiration in local search for sat. In: AAAI
    Google ScholarLocate open access versionFindings
  • Cai S, Su K (2013) Local search for boolean satisfiability with configuration checking and subscore. Artif Intell 204:75–98
    Google ScholarLocate open access versionFindings
  • Cai S, Lin J (2016) Fast solving maximum weight clique problem in massive graphs. In: Proceedings of IJCAI 2016, pp 568–574
    Google ScholarLocate open access versionFindings
  • Cai S, Su K, Sattar A (2011) Local search with edge weighting and configuration checking heuristics for minimum vertex cover. Artif Intell 175(9–10):1672–1696
    Google ScholarLocate open access versionFindings
  • Cai S, Su K, Luo C, Sattar A (2013) NuMVC: an efficient local search algorithm for minimum vertex cover. J Artif Intell Res 46:687–716
    Google ScholarLocate open access versionFindings
  • Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382
    Google ScholarLocate open access versionFindings
  • Fan Y, Li N, Li C, Ma Z, Latecki LJ, Su K (2017a) Restart and random walk in local search for maximum vertex weight cliques with evaluations in clustering aggregation. In: Proceedings of international joint conference on artificial intelligence (IJCAI), pp 622–630
    Google ScholarLocate open access versionFindings
  • Fan Y, Ma Z, Su K, Li C, Rao C, Liu RH, Latecki L (2017b) A local search algorithm for the maximum weight clique problem in large graphs. In: 29rd IEEE international conference on tools with artificial intelligence (ICTAI) 2017. IEEE, pp 1099–1104
    Google ScholarLocate open access versionFindings
  • Fang Z, Li CM, Qiao K, Feng X, Xu K (2014) Solving maximum weight clique using maximum satisfiability reasoning. In: Proceedings of the twenty-first European conference on artificial intelligence. IOS Press, pp 303–308
    Google ScholarLocate open access versionFindings
  • Fomeni FD (2017) A new family of facet defining inequalities for the maximum edge-weighted clique problem. Optim Lett 11(1):47–54
    Google ScholarLocate open access versionFindings
  • Gouveia L, Martins P (2015) Solving the maximum edge-weight clique problem in sparse graphs with compact formulations. EURO J Comput Optim 3(1):1–30
    Google ScholarLocate open access versionFindings
  • Jiang H, Li CM, Manya F (2017) An exact algorithm for the maximum weight clique problem in large graphs. In: AAAI, pp 830–838
    Google ScholarFindings
  • Shimizu S, Yamaguchi K, Masuda S (2018) A branch-and-bound based exact algorithm for the maximum edge-weight clique problem. In: International conference on computational science/intelligence & applied informatics. Springer, pp 27–47
    Google ScholarLocate open access versionFindings
  • 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
    Google ScholarLocate open access versionFindings
  • Tomita E, Seki T (2003) An efficient branch-and-bound algorithm for finding a maximum clique. In: International conference on discrete mathematics and theoretical computer science, pp 278–289
    Google ScholarLocate open access versionFindings
  • Wang Y, Cai S, Yin M (2016) Two efficient local search algorithms for maximum weight clique problem. In: Proceedings of AAAI 2016, pp 805–811
    Google ScholarLocate open access versionFindings
  • Wu Q, Hao JK, Glover F (2012) Multi-neighborhood tabu search for the maximum weight clique problem. Ann Oper Res 196(1):611–634
    Google ScholarLocate open access versionFindings
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