Coloring Graphs Having Few Colorings Over Path Decompositions

Author Andreas Björklund

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Andreas Björklund

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Andreas Björklund. Coloring Graphs Having Few Colorings Over Path Decompositions. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 13:1-13:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them. We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases.
  • Graph vertex coloring
  • path decomposition
  • Alon-Tarsi theorem


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  1. A. Abboud and V. Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proceedings of the IEEE FOCS, pages 434-443, 2014. Google Scholar
  2. N. Alon and M. Tarsi. Colorings and orientations of graphs. Combinatorica, 12:125-134, 1992. Google Scholar
  3. A. Backurs and P. Indyk. Edit distance cannot be computed in strongly subquadratic time (unless seth is false). In Proceedings of the ACM STOC, pages 51-58, 2015. Google Scholar
  4. R. Barbanchon. On unique graph 3-colorability and parsimonious reductions in the plane. Theoretical Computer Science, 319:455-482, 2004. Google Scholar
  5. A. Björklund, H. Dell, and T. Husfeldt. The parity of set systems under random restrictions with applications to exponential time problems. In Proceedings of ICALP, pages 231-242, 2015. Google Scholar
  6. R. L. Brooks. On colouring the nodes of a network. Proc. Cambridge Philosophical Society, Math. Phys. Sci, 37:194-197, 1941. Google Scholar
  7. C. Calabro, R. Impagliazzo, V. Kabanets, and R. Paturi. The complexity of unique k-sat: An isolation lemma for k-cnfs. Journal of Computer and System Sciences, 74:386-393, 2008. Google Scholar
  8. M. Cygan, S. Kratsch, and J. Nederlof. Fast hamiltonicity checking via bases of perfect matchings. In Proceedings of the ACM STOC, pages 301-310, 2013. Google Scholar
  9. D. Hefetz. On two generalizations of the alon-tarsi polynomial method. Journal of Combinatorial Theory Series B, 101:403-414, 2011. Google Scholar
  10. C. Hillar and T. Windfeldt. An algebraic characterization of uniquely vertex colorable graphs. Journal of Combinatorial Theory Series B, 98:400-414, 2008. Google Scholar
  11. I. Holyer. The np-completeness of edge-coloring. SIAM J. Comput., 10:718-720, 1981. Google Scholar
  12. R. Impagliazzo and R. Paturi. The complexity of k-sat. In Proceedings of the IEEE Computational Complexity Conference, pages 237-240, 1999. Google Scholar
  13. T. B. Jensen and B. Toft. Graph Coloring Problems. Wiley-Interscience, 1st edition, 1994. Google Scholar
  14. R. M. Karp. Reducibility among combinatorial problems. Complexity of Computer Computations, pages 85-103, 1972. Google Scholar
  15. D. Lokshtanov, D. Marx, and S. Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. In Proceedings of ACM-SIAM SODA, pages 777-789, 2011. Google Scholar
  16. B. Reed. A strengthening of brook’s theorem. Journal of Combinatorial Theory Series B, 76:136-149, 1999. Google Scholar
  17. N. Robertson and P. Seymour. Graph minors. i. excluding a forest. Journal of Combinatorial Theory Series B, 35:39-61, 1983. Google Scholar
  18. S. Xu. The size of uniquely colorable graphs. Journal of Combinatorial Theory Series B, 50:319-320, 1990. Google Scholar
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