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Faster Graph Coloring in Polynomial Space

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Computing and Combinatorics (COCOON 2017)

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Abstract

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time \(O(1.1389^n)\) for graphs with maximum degree 3 and in time \(O(1.2356^n)\) for general graphs, where n is the number of vertices. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time \(O(2.2356^n)\). As a byproduct, we also obtain an exponential-space \(O(1.2330^n)\) time algorithm for counting independent sets.

Our main algorithm counts independent sets in graphs with maximum degree 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties. The main complication is to upper bound the number of times the algorithm has to branch on vertices all of whose neighbors have degree 2, while still decreasing the size of the separator each time the algorithm branches.

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References

  1. Angelsmark, O., Thapper, J.: Partitioning based algorithms for some colouring problems. In: Hnich, B., Carlsson, M., Fages, F., Rossi, F. (eds.) CSCLP 2005. LNCS (LNAI), vol. 3978, pp. 44–58. Springer, Heidelberg (2006). doi:10.1007/11754602_4

    Chapter  Google Scholar 

  2. Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 575–582. IEEE Computer Society (2006)

    Google Scholar 

  3. Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. 39(2), 546–563 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithm. 21(2), 358–402 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bodlaender, H.L., Kratsch, D.: An exact algorithm for graph coloring with polynomial memory. Technical report UU-CS-2006-015, Department of Information and Computing Sciences, Utrecht University (2006)

    Google Scholar 

  7. Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Oper. Res. Lett. 32(6), 547–556 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Christofides, N.: An algorithm for the chromatic number of a graph. Comput. J. 14(1), 38–39 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dahllöf, V., Jonsson, P., Wahlström, M.: Counting models for 2SAT and 3SAT formulae. Theor. Comput. Sci. 332(1–3), 265–291 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eppstein, D.: Small maximal independent sets and faster exact graph coloring. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 462–470. Springer, Heidelberg (2001). doi:10.1007/3-540-44634-6_42

    Chapter  Google Scholar 

  11. Eppstein, D.: Small maximal independent sets and faster exact graph coloring. J. Graph Algorithms Appl. 7(2), 131–140 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feder, T., Motwani, R.: Worst-case time bounds for coloring and satisfiability problems. J. Algorithm. 45(2), 192–201 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. J. ACM 56(5), 25 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fomin, F.V., Høie, K.: Pathwidth of cubic graphs and exact algorithms. Inform. Process. Lett. 97(5), 191–196 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Springer Science & Business Media, New York (2010)

    Book  MATH  Google Scholar 

  17. Fürer, M., Kasiviswanathan, S.P.: Algorithms for counting 2-Sat solutions and colorings with applications. In: Kao, M.-Y., Li, X.-Y. (eds.) AAIM 2007. LNCS, vol. 4508, pp. 47–57. Springer, Heidelberg (2007). doi:10.1007/978-3-540-72870-2_5

    Chapter  Google Scholar 

  18. Gaspers, S.: Exponential Time Algorithms. VDM Verlag, Saarbrücken (2010)

    MATH  Google Scholar 

  19. Gaspers, S., Lee, E.: Faster graph coloring in polynomial space. CoRR, abs/1607.06201 (2016)

    Google Scholar 

  20. Gaspers, S., Sorkin, G.B.: A universally fastest algorithm for Max 2-Sat, Max 2-CSP, and everything in between. J. Comput. Syst. Sci. 78(1), 305–335 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gaspers, S., Sorkin, G.B.: Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 567–579. Springer, Heidelberg (2015). doi:10.1007/978-3-662-47672-7_46

    Chapter  Google Scholar 

  22. Greenhill, C.: The complexity of counting colourings and independent sets in sparse graphs and hypergraphs. Comput. Complex. 9(1), 52–72 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Iwata, Y.: A faster algorithm for dominating set analyzed by the potential method. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 41–54. Springer, Heidelberg (2012). doi:10.1007/978-3-642-28050-4_4

    Chapter  Google Scholar 

  24. Junosza-Szaniawski, K., Tuczynski, M.: Counting independent sets via divide measure and conquer method. Technical report abs/1503.08323, arXiv CoRR (2015)

    Google Scholar 

  25. Koivisto, M.: An \({O}^*(2^n)\) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 583–590. IEEE Computer Society (2006)

    Google Scholar 

  26. Lawler, E.L.: A note on the complexity of the chromatic number problem. Inform. Process. Lett. 5(3), 66–67 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. Roth, D.: On the hardness of approximate reasoning. Artif. Intell. 82(1), 273–302 (1996)

    Article  MathSciNet  Google Scholar 

  28. Scott, A.D., Sorkin, G.B.: Polynomial constraint satisfaction problems, graph bisection, and the ising partition function. ACM T. Algorithms 5(4), 45 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Wahlström, M.: Exact algorithms for finding minimum transversals in rank-3 hypergraphs. J. Algorithm 51(2), 107–121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wahlström, M.: A tighter bound for counting max-weight solutions to 2SAT instances. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 202–213. Springer, Heidelberg (2008). doi:10.1007/978-3-540-79723-4_19

    Chapter  Google Scholar 

  31. Woeginger, G.J.: Open problems around exact algorithms. Discret. Appl. Math. 156(3), 397–405 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Magnus Wahlström for clarifying an issue of the case analysis in [30] and an anonymous reviewer for useful comments on an earlier version of the paper. Serge Gaspers is the recipient of an Australian Research Council (ARC) Future Fellowship (FT140100048) and acknowledges support under the ARC’s Discovery Projects funding scheme (DP150101134).

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Authors and Affiliations

  1. UNSW, Sydney, Australia

    Serge Gaspers & Edward J. Lee

  2. Data61, CSIRO, Sydney, Australia

    Serge Gaspers & Edward J. Lee

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  1. Serge Gaspers
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  2. Edward J. Lee
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Correspondence to Edward J. Lee .

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Editors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Texas A&M University, College Station, Texas, USA

    Jianer Chen

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Gaspers, S., Lee, E.J. (2017). Faster Graph Coloring in Polynomial Space. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_31

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  • DOI: https://doi.org/10.1007/978-3-319-62389-4_31

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  • Online ISBN: 978-3-319-62389-4

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