Abstract
The graph coloring problem appears in numerous applications, yet many state-of-the-art methods are hardly applicable to real world, very large, networks. The most efficient approaches for massive graphs rely on “peeling” the graph of its low-degree vertices and focus on the maximum k-core where k is some lower bound on the chromatic number of the graph. However, unless the graphs are extremely sparse, the cores can be very large, and lower and upper bounds are often obtained using greedy heuristics.
In this paper, we introduce a combined approach using local search to find good quality solutions on massive graphs as well as locate small subgraphs with potentially large chromatic number. The subgraphs can be used to compute good lower bounds, which makes it possible to solve optimally extremely large graphs, even when they have large k-cores.
G. Katsirelos—The second author was partially supported by the french “Agence nationale de la Recherche”, project DEMOGRAPH, reference ANR-16-C40-0028.
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Notes
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denotes our implementation of the Dsatur heuristic. - 3.
Ties broken by overall degree.
- 4.
Sources available at: https://bitbucket.org/gkatsi/minicsp.
- 5.
Sources available at: https://bitbucket.org/gkatsi/gc-cdcl/src/master/.
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Personnal communication with the authors.
- 7.
email-Eu-core, email-EuAll, Gnutella08/09, bitcoinalpha, bitcoinotc, facebook, gplus, CollegeMsg and sx-superuser.
References
Aardal, K.I., Hoesel, S.P.M.V., Koster, A.M.C.A., Mannino, C., Sassano, A.: Models and solution techniques for frequency assignment problems. Ann. Oper. Res. 153(1), 79–129 (2007)
Abello, J., Pardalos, P., Resende, M.G.C.: On maximum clique problems in very large graphs. In: Abello, J.M., Vitter, J.S. (eds.) External Memory Algorithms, pp. 119–130. American Mathematical Society, Boston (1999)
Bader, D.A., Meyerhenke, H., Sanders, P., Wagner, D.: Graph partitioning and graph clustering (2012). http://www.cc.gatech.edu/dimacs10/
Blöchliger, I., Zufferey, N.: A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comput. Oper. Res. 35(3), 960–975 (2008)
Brélaz, D.: New methods to color the vertices of a graph. Commun. ACM 22(4), 251–256 (1979)
Chaitin, G.J., Auslander, M.A., Chandra, A.K., Cocke, J., Hopkins, M.E., Markstein, P.W.: Register allocation via coloring. Comput. Lang. 6(1), 47–57 (1981)
Cornaz, D., Jost, V.: A one-to-one correspondence between colorings and stable sets. Oper. Res. Lett. 36(6), 673–676 (2008)
de Werra, D.: An introduction to timetabling. Eur. J. Oper. Res. 19(2), 151–162 (1985)
Furini, F., Gabrel, V., Ternier, I.-C.: Lower bounding techniques for DSATUR-based branch and bound. Electron. Notes Discrete Math. 52, 149–156 (2016). INOC 2015–7th International Network Optimization Conference
Hao, J.-K., Qinghua, W.: Improving the extraction and expansion method for large graph coloring. Discrete Appl. Math. 160(16–17), 2397–2407 (2012)
Hebrard, E., Katsirelos, G.: Clause learning and new bounds for graph coloring. In: Hooker, J. (ed.) CP 2018. LNCS, vol. 11008, pp. 179–194. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98334-9_12
Hertz, A., de Werra, D.: Using tabu search techniques for graph coloring. Computing 39(4), 345–351 (1987)
Johnson, D.J., Trick, M.A. (eds.): Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, Workshop, October 11–13, 1993. American Mathematical Society, Boston (1996)
Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84, 489–506 (1979)
Leskovec, J., Krevl, A.: SNAP datasets: Stanford large network dataset collection (2014). http://snap.stanford.edu/data
Lin, J., Cai, S., Luo, C., Su, K.: A reduction based method for coloring very large graphs. In: Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, pp. 517–523 (2017)
Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theor. 25(1), 1–7 (2006)
Malaguti, E., Monaci, M., Toth, P.: An exact approach for the vertex coloring problem. Discrete Optim. 8(2), 174–190 (2011)
Matula, D.W., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. ACM 30(3), 417–427 (1983)
Mehrotra, A., Trick, M.A.: A column generation approach for graph coloring. INFORMS J. Comput. 8, 344–354 (1995)
Moalic, L., Gondran, A.: Variations on memetic algorithms for graph coloring problems. CoRR, abs/1401.2184 (2014)
Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)
Park, T., Lee, C.Y.: Application of the graph coloring algorithm to the frequency assignment problem. J. Oper. Res. Soc. Jpn. 39(2), 258–265 (1996)
Rossi, R.A., Ahmed, N.K.: Coloring large complex networks. CoRR, abs/1403.3448 (2014)
Schaafsma, B., Heule, M.J.H., van Maaren, H.: Dynamic symmetry breaking by simulating zykov contraction. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 223–236. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02777-2_22
Segundo, P.S.: A new DSATUR-based algorithm for exact vertex coloring. Comput. Oper. Res. 39(7), 1724–1733 (2012)
Van Gelder, A.: Another look at graph coloring via propositional satisfiability. Discrete Appl. Math. 156(2), 230–243 (2008)
Verma, A., Buchanan, A., Butenko, S.: Solving the maximum clique and vertex coloring problems on very large sparse networks. INFORMS J. Comput. 27(1), 164–177 (2015)
Walteros, J.L., Buchanan, A.: Why is maximum clique often easy in practice? Optimization Online (2018). http://www.optimization-online.org/DB_HTML/2018/07/6710.html
Zhou, Z., Li, C.-M., Huang, C., Ruchu, X.: An exact algorithm with learning for the graph coloring problem. Comput. Oper. Res. 51, 282–301 (2014)
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Hebrard, E., Katsirelos, G. (2019). A Hybrid Approach for Exact Coloring of Massive Graphs. In: Rousseau, LM., Stergiou, K. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2019. Lecture Notes in Computer Science(), vol 11494. Springer, Cham. https://doi.org/10.1007/978-3-030-19212-9_25
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