Document Open Access Logo

Finer Tight Bounds for Coloring on Clique-Width

Author Michael Lampis

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2018.86.pdf
  • Filesize: 454 kB
  • 14 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Clique-width
  • SETH
  • Coloring

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We revisit the complexity of the classical k-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors k is large. However, much less is known on its complexity for small, concrete values of k. In this paper, we completely determine the complexity of k-Coloring parameterized by clique-width for any fixed k, under the SETH. Specifically, we show that for all k >= 3,epsilon>0, k-Coloring cannot be solved in time O^*((2^k-2-epsilon)^{cw}), and give an algorithm running in time O^*((2^k-2)^{cw}). Thus, if the SETH is true, 2^k-2 is the "correct" base of the exponent for every k.
Along the way, we also consider the complexity of k-Coloring parameterized by the related parameter modular treewidth (mtw). In this case we show that the "correct" running time, under the SETH, is O^*({k choose floor[k/2]}^{mtw}). If we base our results on a weaker assumption (the ETH), they imply that k-Coloring cannot be solved in time n^{o(cw)}, even on instances with O(log n) colors.

Cite As Get BibTex

Michael Lampis. Finer Tight Bounds for Coloring on Clique-Width. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 86:1-86:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.86

Author Details

Michael Lampis
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 , LAMSADE, 75016, Paris, France

References

  1. Hans L. Bodlaender, Erik Jan van Leeuwen, Johan M. M. van Rooij, and Martin Vatshelle. Faster algorithms on branch and clique decompositions. In MFCS, volume 6281 of Lecture Notes in Computer Science, pages 174-185. Springer, 2010. Google Scholar
  2. Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna. Tight lower bounds for the complexity of multicoloring. In ESA, volume 87 of LIPIcs, pages 18:1-18:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  3. Glencora Borradaile and Hung Le. Optimal dynamic program for r-domination problems over tree decompositions. In IPEC, volume 63 of LIPIcs, pages 8:1-8:23. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  4. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, volume 138 of Encyclopedia of mathematics and its applications. Cambridge University Press, 2012. Google Scholar
  5. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. Google Scholar
  6. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. Google Scholar
  7. Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On problems as hard as CNF-SAT. ACM Trans. Algorithms, 12(3):41:1-41:24, 2016. Google Scholar
  8. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
  9. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In FOCS, pages 150-159. IEEE Computer Society, 2011. Google Scholar
  10. David P. Dailey. Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete. Discrete Mathematics, 30(3):289-293, 1980. Google Scholar
  11. Uriel Feige and Joe Kilian. Zero knowledge and the chromatic number. J. Comput. Syst. Sci., 57(2):187-199, 1998. Google Scholar
  12. Michael R. Fellows, Frances A. Rosamond, Udi Rotics, and Stefan Szeider. Clique-width is np-complete. SIAM J. Discrete Math., 23(2):909-939, 2009. Google Scholar
  13. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. Google Scholar
  14. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Intractability of clique-width parameterizations. SIAM J. Comput., 39(5):1941-1956, 2010. Google Scholar
  15. Jakub Gajarský, Michael Lampis, and Sebastian Ordyniak. Parameterized algorithms for modular-width. In IPEC, volume 8246 of Lecture Notes in Computer Science, pages 163-176. Springer, 2013. Google Scholar
  16. Robert Ganian. Twin-cover: Beyond vertex cover in parameterized algorithmics. In IPEC, volume 7112 of Lecture Notes in Computer Science, pages 259-271. Springer, 2011. Google Scholar
  17. Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Cliquewidth III: the odd case of graph coloring parameterized by cliquewidth. In SODA, pages 262-273. SIAM, 2018. Google Scholar
  18. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. Google Scholar
  19. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Google Scholar
  20. Lars Jaffke and Bart M. P. Jansen. Fine-grained parameterized complexity analysis of graph coloring problems. In CIAC, volume 10236 of Lecture Notes in Computer Science, pages 345-356, 2017. Google Scholar
  21. Daniel Kobler and Udi Rotics. Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics, 126(2-3):197-221, 2003. Google Scholar
  22. Michael Lampis. Algorithmic meta-theorems for restrictions of treewidth. Algorithmica, 64(1):19-37, 2012. URL: http://dx.doi.org/10.1007/s00453-011-9554-x.
  23. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs on bounded treewidth are probably optimal. In SODA, pages 777-789. SIAM, 2011. Google Scholar
  24. Dániel Marx and Valia Mitsou. Double-exponential and triple-exponential bounds for choosability problems parameterized by treewidth. In ICALP, volume 55 of LIPIcs, pages 28:1-28:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  25. Stefan Mengel. Parameterized compilation lower bounds for restricted cnf-formulas. In SAT, volume 9710 of Lecture Notes in Computer Science, pages 3-12. Springer, 2016. Google Scholar
  26. Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width. J. Comb. Theory, Ser. B, 96(4):514-528, 2006. Google Scholar
  27. Daniël Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model counting for CNF formulas of bounded modular treewidth. Algorithmica, 76(1):168-194, 2016. Google Scholar
  28. Sigve Hortemo Sæther and Jan Arne Telle. Between treewidth and clique-width. Algorithmica, 75(1):218-253, 2016. Google Scholar
  29. Johan M. M. van Rooij, Hans L. Bodlaender, and Peter Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In ESA, volume 5757 of Lecture Notes in Computer Science, pages 566-577. Springer, 2009. Google Scholar
  30. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3(1):103-128, 2007. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact