Abstract
We introduce a generic algorithmic technique and apply it on decision and counting versions of graph coloring. Our approach is based on the following idea: either a graph has nice (from the algorithmic point of view) properties which allow a simple recursive procedure to find the solution fast, or the pathwidth of the graph is small, which in turn can be used to find the solution by dynamic programming. By making use of this technique we obtain the fastest known exact algorithms
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running in time O(1.7272n) for deciding if a graph is 4-colorable
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running in time O(1.6262n) and O(1.9464n) for counting the number of k-colorings for k = 3 and 4 respectively.
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References
Angelsmark, O., Jonsson, P.: Improved Algorithms for Counting Solutions in Constraint Satisfaction Problems. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 81–95. Springer, Heidelberg (2003)
Beigel, R., Eppstein, D.: 3-coloring in time O(1.3289n).. Journal of Algorithms 54(2), 168–204 (2005)
Björklund, A., Husfeldt, T.: Inclusion–Exclusion Algorithms for Counting Set Partitions. In: The Proceedings of FOCS 2006, pp. 575–582 (2006)
Byskov, J.M.: Exact Algorithms for Graph Colouring and Exact Satisfiability. PhD Dissertation (2004).
Byskov, J.M.: Enumerating Maximal Independent Sets with Applications to Graph Colouring. Operations Research Letters 32(6), 547–556 (2004)
Christofides, N.: An algorithm for the chromatic number of a graph. Computer J. 14, 38–39 (1971)
Eppstein, D.: Small Maximal Independent Sets and Faster Exact Graph Coloring. J. Graph Algorithms Appl. 7(2), 131–140 (2003)
Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On Two Techniques of Combining Branching and Treewidth. Report No 337, Department of Informatics, University of Bergen, Norway (December 2006)
Fomin, F.V., Høie, K.: Pathwidth of cubic graphs and exact algorithms. Information Processing Letters 97(5), 191–196 (2006)
Feige, U., Kilian, J.: Zero Knowledge and the Chromatic Number. Journal of Computer and System Sciences 57(2), 187–199 (1998)
Fürer, M., Kasiviswanathan, S.P.: Algorithms for counting 2-SAT solutions and colorings with applications. In: ECCC 33 (2005)
Garey, M.R., Johnson, D.S.: Computer and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco, CA (1979)
Khot, S., Ponnuswami, A.K.: Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 226–237. Springer, Heidelberg (2006)
Koivisto, M.: An O(2n) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: Proceedings of FOCS 2006, pp. 583–590 (2006)
Lawler, E.L.: A Note on the Complexity of the Chromatic Number. Information Processing Letters 5(3), 66–67 (1976)
Monien, B., Preis, R.: Upper bounds on the bisection width of 3- and 4-regular graphs. Discrete Algorithms 4(3), 475–498 (2006)
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Fomin, F.V., Gaspers, S., Saurabh, S. (2007). Improved Exact Algorithms for Counting 3- and 4-Colorings. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_9
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DOI: https://doi.org/10.1007/978-3-540-73545-8_9
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