Reduction and Local Search for Weighted Graph Coloring Problem

national conference on artificial intelligence, 2020.

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maximum weight clique problemconfiguration checkingconventional benchmarkvertex coloring problemmaximum weight stable set problemMore(10+)
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This paper introduced a lower bound, a reduction rule, and a local search algorithm for weighted graph coloring problem

Abstract:

The weighted graph coloring problem (WGCP) is an important extension of the graph coloring problem (GCP) with wide applications. Compared to GCP, where numerous methods have been developed and even massive graphs with millions of vertices can be solved well, fewer works have been done for WGCP, and no solution is available for solving WGC...More

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Introduction
  • In classical GCP, a basic assumption is that vertices in the graph are important, it is hard to hold in many real world scenarios where each vertex is associated with various types of weights.
  • WGCP aims to partition all the vertices into several disjoint subsets such that the sum of those subset costs is minimized, where the cost of each subset is given by the maximum weight of a vertex within the current subset.
  • WGCP can be directly encoded as the maximum weight stable set problem (MWSS) (Cornaz, Furini, and Malaguti 2017)
Highlights
  • Graph coloring problem (GCP) is a well-known combinatorial optimization problem
  • Motivated to contribute to solving weighted graph coloring problem (WGCP) problems with massive graphs, in this paper we propose a novel efficient WGCP algorithm, namely RedLS (Reduction plus Local Search)
  • RedLS proves the optimal solution for 4 instances from the conventional benchmarks, and the number of reduced vertices is on average 12.45, indicating that the reduction rule is not effective for conventional benchmarks
  • This paper introduced a lower bound, a reduction rule, and a local search algorithm for WGCP
  • We proposed the reduction rule based on clique sampling to remove some unnecessary vertices
  • Experiments on conventional benchmarks and massive graphs indicate that RedLS significantly outperforms the state-of-the-art algorithms
Results
  • RedLS finds better values than 2Phase for 96 instances.
  • The authors mainly compare RedLS with MWSS and AFISA.
  • Most instances are so easy that RedLS, MWSS and AFISA find the same quality values.
  • The authors do not report the detailed results of such instances in Table 1, but the authors summarize the run time comparisons in Figure 1.
  • The good results of RedLS on conventional benchmarks mainly come from the power of the underlying local search algorithm
Conclusion
  • This paper introduced a lower bound, a reduction rule, and a local search algorithm for WGCP.
  • The authors proposed the reduction rule based on clique sampling to remove some unnecessary vertices.
  • In the local search algorithm, the authors designed the selection rules and the new variant of configuration checking to determine which operation is the candidate selected operation in the local search procedure.
  • As for future work, the authors will attempt to further improve RedLS via a few novel reduction rules
Summary
  • Introduction:

    In classical GCP, a basic assumption is that vertices in the graph are important, it is hard to hold in many real world scenarios where each vertex is associated with various types of weights.
  • WGCP aims to partition all the vertices into several disjoint subsets such that the sum of those subset costs is minimized, where the cost of each subset is given by the maximum weight of a vertex within the current subset.
  • WGCP can be directly encoded as the maximum weight stable set problem (MWSS) (Cornaz, Furini, and Malaguti 2017)
  • Results:

    RedLS finds better values than 2Phase for 96 instances.
  • The authors mainly compare RedLS with MWSS and AFISA.
  • Most instances are so easy that RedLS, MWSS and AFISA find the same quality values.
  • The authors do not report the detailed results of such instances in Table 1, but the authors summarize the run time comparisons in Figure 1.
  • The good results of RedLS on conventional benchmarks mainly come from the power of the underlying local search algorithm
  • Conclusion:

    This paper introduced a lower bound, a reduction rule, and a local search algorithm for WGCP.
  • The authors proposed the reduction rule based on clique sampling to remove some unnecessary vertices.
  • In the local search algorithm, the authors designed the selection rules and the new variant of configuration checking to determine which operation is the candidate selected operation in the local search procedure.
  • As for future work, the authors will attempt to further improve RedLS via a few novel reduction rules
Tables
  • Table1: Results of RedLS, AFISA, MWSS and 2Phase on conventional benchmarks
  • Table2: Results of RedLS and AFISA on massive graphs with w1
  • Table3: Summary of comparison between RedLS, AFISA, MWSS and 2Phase on massive graph with w2. #Better indicates the number of instances where an algorithm finds better minimal (average) solutions. #N/A denotes the number of instances where an algorithm fails to find a solution under the given time limit
Download tables as Excel
Funding
  • This work is supported by the Fundamental Research Funds for the Central Universities 2412018ZD017, NSFC (under grant nos. 61806050, 61972063, 61976050, 61972384)
  • Shaowei Cai was supported by Youth Innovation Promotion Association, Chinese Academy of Sciences (No.2017150)
Reference
  • Cai, S., and Lin, J. 2016. Fast solving maximum weight clique problem in massive graphs. In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, 568–574.
    Google ScholarLocate open access versionFindings
  • Cai, S.; Su, K.; and Sattar, A. 2011. Local search with edge weighting and configuration checking heuristics for minimum vertex cover. Artificial Intelligence 175(9-10):1672–1696.
    Google ScholarLocate open access versionFindings
  • Cai, S. 2015. Balance between complexity and quality: Local search for minimum vertex cover in massive graphs. In Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, 747–753.
    Google ScholarLocate open access versionFindings
  • Cornaz, D.; Furini, F.; and Malaguti, E. 2017. Solving vertex coloring problems as maximum weight stable set problems. Discrete Applied Mathematics 217:151–162.
    Google ScholarLocate open access versionFindings
  • Furini, F., and Malaguti, E. 2012. Exact weighted vertex coloring via branch-and-price. Discrete Optimization 9(2):130–136.
    Google ScholarLocate open access versionFindings
  • Garey, M. R., and Johnson, D. S. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman.
    Google ScholarFindings
  • Gavranovic, H., and Finke, G. 2000. Graph partitioning and set covering for the optimal design of a production system in the metal industry. IFAC Proceedings Volumes 33(17):603–608.
    Google ScholarLocate open access versionFindings
  • Glover, F. 1989. Tabu search-part i. ORSA Journal on computing 1(3):190–206.
    Google ScholarLocate open access versionFindings
  • Hebrard, E., and Katsirelos, G. 2018. Clause learning and new bounds for graph coloring. In International Conference on Principles and Practice of Constraint Programming, 179–194.
    Google ScholarLocate open access versionFindings
  • Hebrard, E., and Katsirelos, G. 2019. A hybrid approach for exact coloring of massive graphs. In International Conference on Integration of Constraint Programming, Artificial Intelligence, and Operations Research, 374–390.
    Google ScholarLocate open access versionFindings
  • Hochbaum, D. S., and Landy, D. 1997. Scheduling semiconductor burn-in operations to minimize total flowtime. Operations research 45(6):874–885.
    Google ScholarLocate open access versionFindings
  • Hsu, H.-C., and Chang, G. J. 2016. Max-coloring of vertexweighted graphs. Graphs and Combinatorics 32(1):191–198.
    Google ScholarLocate open access versionFindings
  • Lin, J.; Cai, S.; Luo, C.; and Su, K. 2017. A reduction based method for coloring very large graphs. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, 517–523.
    Google ScholarLocate open access versionFindings
  • Malaguti, E.; Monaci, M.; and Toth, P. 2009. Models and heuristic algorithms for a weighted vertex coloring problem. Journal of Heuristics 15(5):503–526.
    Google ScholarLocate open access versionFindings
  • McCreesh, C.; Prosser, P.; Simpson, K.; and Trimble, J. 2017. On maximum weight clique algorithms, and how they are evaluated. In International Conference on Principles and Practice of Constraint Programming, 206–225.
    Google ScholarLocate open access versionFindings
  • Pemmaraju, S. V.; Raman, R.; and Varadarajan, K. 2004. Buffer minimization using max-coloring. In Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, 562–571.
    Google ScholarLocate open access versionFindings
  • Peng, Y.; Choi, B.; He, B.; Zhou, S.; Xu, R.; and Yu, X. 2016. Vcolor: A practical vertex-cut based approach for coloring large graphs. In Data Engineering (ICDE), 2016 IEEE 32nd International Conference on, 97–108.
    Google ScholarLocate open access versionFindings
  • Prais, M., and Ribeiro, C. C. 2000. Reactive grasp: An application to a matrix decomposition problem in tdma traffic assignment. INFORMS Journal on Computing 12(3):164–176.
    Google ScholarLocate open access versionFindings
  • Ribeiro, C. C.; Minoux, M.; and Penna, M. C. 1989. An optimal column-generation-with-ranking algorithm for very large scale set partitioning problems in traffic assignment. European Journal of Operational Research 41(2):232–239.
    Google ScholarLocate open access versionFindings
  • Rossi, R. A., and Ahmed, N. K. 2015. The network data repository with interactive graph analytics and visualization. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, 4292–4293.
    Google ScholarLocate open access versionFindings
  • Sun, W.; Hao, J.-K.; Lai, X.; and Wu, Q. 2018. Adaptive feasible and infeasible tabu search for weighted vertex coloring. Information Sciences 466:203–219.
    Google ScholarLocate open access versionFindings
  • Verma, A.; Buchanan, A.; and Butenko, S. 2015. Solving the maximum clique and vertex coloring problems on very large sparse networks. INFORMS Journal on computing 27(1):164–177.
    Google ScholarLocate open access versionFindings
  • Wang, Y.; Cai, S.; Chen, J.; and Yin, M. 2018. A fast local search algorithm for minimum weight dominating set problem on massive graphs. In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, 1514–1522.
    Google ScholarLocate open access versionFindings
  • Wang, Y.; Cai, S.; and Yin, M. 2016. Two efficient local search algorithms for maximum weight clique problem. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 805– 811.
    Google ScholarLocate open access versionFindings
  • Wang, Y.; Cai, S.; and Yin, M. 2017. Local search for minimum weight dominating set with two-level configuration checking and frequency based scoring function. Journal of Artificial Intelligence Research 58:267–295.
    Google ScholarLocate open access versionFindings
  • Zhou, Y.; Duval, B.; and Hao, J.-K. 2018. Improving probability learning based local search for graph coloring. Applied Soft Computing 65:542–553.
    Google ScholarLocate open access versionFindings
  • Zuckerman, D. 2006. Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, 681–690.
    Google ScholarLocate open access versionFindings
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