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The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width

Authors Robert Ganian , Thekla Hamm , Viktoriia Korchemna, Karolina Okrasa , Kirill Simonov

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Author Details

Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Thekla Hamm
  • Algorithms and Complexity Group, TU Wien, Austria
Viktoriia Korchemna
  • Algorithms and Complexity Group, TU Wien, Austria
Karolina Okrasa
  • Faculty of Matematics and Information Science, Warsaw University of Technology, Poland
  • Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Kirill Simonov
  • Algorithms and Complexity Group, TU Wien, Austria

Funding

  • Ganian, Robert: Robert Ganian acknowledges support by the Austrian Science Fund (FWF, projects Y1329 and P31336).
  • Hamm, Thekla: Thekla Hamm acknowledges support by the Austrian Science Fund (FWF, projects P31336 and Y1329).
  • Korchemna, Viktoriia: Viktoriia Korchemna acknowledges support by the Austrian Science Fund (FWF, project Y1329).
  • Okrasa, Karolina: Karolina Okrasa acknowledges support by the European Research Council, grant agreement No 714704. Parts of this work were performed while visiting TU Wien, Vienna, Austria.
  • Simonov, Kirill: Kirill Simonov acknowledges support by the Austrian Science Fund (FWF, project P31336).

Cite AsGet BibTex

Robert Ganian, Thekla Hamm, Viktoriia Korchemna, Karolina Okrasa, and Kirill Simonov. The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 66:1-66:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.66

Abstract

The generic homomorphism problem, which asks whether an input graph G admits a homomorphism into a fixed target graph H, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of G (denoted cw) for virtually all choices of H under the Strong Exponential Time Hypothesis. In particular, we identify a property of H called the signature number s(H) and show that for each H, the homomorphism problem can be solved in time O^*(s(H)^cw). Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each H that is either a projective core or a graph admitting a factorization with additional properties - allowing us to cover all possible target graphs under long-standing conjectures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • homomorphism
  • clique-width
  • fine-grained complexity

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References

  1. Rémy Belmonte, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Yota Otachi. Grundy distinguishes treewidth from pathwidth. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 14:1-14:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  2. Jan Böker. Graph similarity and homomorphism densities. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 32:1-32:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  3. Andrei A. Bulatov and Amineh Dadsetan. Counting homomorphisms in plain exponential time. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 21:1-21:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  4. 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
  5. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. Google Scholar
  6. Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socała. Tight lower bounds on graph embedding problems. J. ACM, 64(3):18:1-18:22, 2017. URL: https://doi.org/10.1145/3051094.
  7. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  8. Willibald Dörfler. Primfaktorzerlegung und Automorphismen des Kardinalproduktes von Graphen. Glasnik Matematički, 9:15-27, 1974. Google Scholar
  9. László Egri, Dániel Marx, and Paweł Rzążewski. Finding list homomorphisms from bounded-treewidth graphs to reflexive graphs: a complete complexity characterization. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, pages 27:1-27:15, 2018. Google Scholar
  10. Tomás Feder, Pavol Hell, and Jing Huang. Bi-arc graphs and the complexity of list homomorphisms. J. Graph Theory, 42(1):61-80, 2003. URL: https://doi.org/10.1002/jgt.10073.
  11. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput., 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  12. Fedor V. Fomin, Pinar Heggernes, and Dieter Kratsch. Exact algorithms for graph homomorphisms. Theory Comput. Syst., 41(2):381-393, 2007. URL: https://doi.org/10.1007/s00224-007-2007-x.
  13. Martin Grohe. The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM, 54(1):1:1-1:24, 2007. Google Scholar
  14. Richard H. Hammack, Wilfried Imrich, and Sandi Klavžar. Handbook of product graphs. CRC press, 2011. Google Scholar
  15. Pavol Hell and Jaroslav Nešetřil. On the complexity of H-coloring. J. Comb. Theory, Ser. B, 48(1):92-110, 1990. URL: https://doi.org/10.1016/0095-8956(90)90132-J.
  16. Pavol Hell and Jaroslav Nešetřil. The core of a graph. Discret. Math., 109(1-3):117-126, 1992. Google Scholar
  17. Sang il Oum and Paul Seymour. Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B, 96(4):514-528, 2006. URL: https://doi.org/10.1016/j.jctb.2005.10.006.
  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. URL: https://doi.org/10.1006/jcss.2001.1774.
  20. Michael Lampis. Finer tight bounds for coloring on clique-width. SIAM J. Discret. Math., 34(3):1538-1558, 2020. URL: https://doi.org/10.1137/19M1280326.
  21. Benoît Larose. Families of strongly projective graphs. Discuss. Math. Graph Theory, 22(2):271-292, 2002. URL: https://doi.org/10.7151/dmgt.1175.
  22. Benoît Larose and Claude Tardif. Strongly rigid graphs and projectivity. Multiple-Valued Logic, 7:339-361, 2001. Google Scholar
  23. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bull. EATCS, 105:41-72, 2011. Google Scholar
  24. Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Full complexity classification of the list homomorphism problem for bounded-treewidth graphs. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms, ESA 2020, September 7-9, 2020, Pisa, Italy (Virtual Conference), volume 173 of LIPIcs, pages 74:1-74:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.74.
  25. Karolina Okrasa and Paweł Rzążewski. Fine-grained complexity of the graph homomorphism problem for bounded-treewidth graphs. SIAM J. Comput., 50(2):487-508, 2021. Google Scholar
  26. Sang-il Oum. Approximating rank-width and clique-width quickly. In Dieter Kratsch, editor, Graph-Theoretic Concepts in Computer Science, pages 49-58, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. Google Scholar
  27. Marta Piecyk and Paweł Rzążewski. Fine-grained complexity of the list homomorphism problem: Feedback vertex set and cutwidth. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021, March 16-19, 2021, Saarbrücken, Germany (Virtual Conference), volume 187 of LIPIcs, pages 56:1-56:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  28. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. Google Scholar
  29. Paweł Rzążewski. Exact algorithm for graph homomorphism and locally injective graph homomorphism. Inf. Process. Lett., 114(7):387-391, 2014. Google Scholar
  30. Magnus Wahlström. New plain-exponential time classes for graph homomorphism. Theory Comput. Syst., 49(2):273-282, 2011. URL: https://doi.org/10.1007/s00224-010-9261-z.
  31. Tomasz Łuczak and Jaroslav Nešetřil. Note on projective graphs. J. Graph Theory, 47(2):81-86, 2004. Google Scholar
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