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Counting Homomorphisms in Plain Exponential Time

Authors Andrei A. Bulatov, Amineh Dadsetan

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • graph homomorphisms
  • plain exponential time
  • clique width

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Abstract

In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the Exponential Time Hypothesis fails there is no algorithm that solves this problem in time O(|V(H)|^o(|V(G)|)). This, however, does not rule out the possibility that faster algorithms exist for restricted problems of this kind. Wahlström proved that #GraphHom can be solved in plain exponential time, that is, in time O((2k+1)^(|V(G)|+|V(H)|) poly(|V(H)|,|V(G)|)) provided H has clique width k. We generalize this result to a larger class of graphs, and also identify several other graph classes that admit a plain exponential algorithm for #GraphHom.

Cite As Get BibTex

Andrei A. Bulatov and Amineh Dadsetan. Counting Homomorphisms in Plain Exponential Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.21

Author Details

Andrei A. Bulatov
  • School of Computing Science, Simon Fraser University, Burnaby, Canada
Amineh Dadsetan
  • School of Computing Science, Simon Fraser University, Burnaby, Canada

Funding

  • Bulatov, Andrei A.: This work was supported by an NSERC Discovery grant.

References

  1. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 684-697, 2016. Google Scholar
  2. Marthe Bonamy, Lukasz Kowalik, Michal Pilipczuk, Arkadiusz Socala, and Marcin Wrochna. Tight lower bounds for the complexity of multicoloring. TOCT, 11(3):13:1-13:19, 2019. Google Scholar
  3. Flavia Bonomo, Luciano N. Grippo, Martin Milanic, and Martín Darío Safe. Graph classes with and without powers of bounded clique-width. Discrete Applied Mathematics, 199:3-15, 2016. Google Scholar
  4. Andreas Brandstädt, Feodor F. Dragan, Hoàng-Oanh Le, and Raffaele Mosca. New graph classes of bounded clique-width. Theory Comput. Syst., 38(5):623-645, 2005. Google Scholar
  5. Jianer Chen, Xiuzhen Huang, Iyad A. Kanj, and Ge Xia. Strong computational lower bounds via parameterized complexity. J. Comput. Syst. Sci., 72(8):1346-1367, 2006. Google Scholar
  6. Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socala. Tight bounds for graph homomorphism and subgraph isomorphism. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1643-1649, 2016. Google Scholar
  7. Amineh Dadsetan and Andrei A. Bulatov. Counting homomorphisms in plain exponential time. CoRR, abs/1810.03087, 2018. URL: http://arxiv.org/abs/1810.03087.
  8. Víctor Dalmau and Peter Jonsson. The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci., 329(1-3):315-323, 2004. Google Scholar
  9. Martin E. Dyer and Catherine S. Greenhill. The complexity of counting graph homomorphisms. Random Struct. Algorithms, 17(3-4):260-289, 2000. Google Scholar
  10. Fedor V. Fomin, Serge Gaspers, and Saket Saurabh. Improved exact algorithms for counting 3- and 4-colorings. In Computing and Combinatorics, 13th Annual International Conference, COCOON 2007, Banff, Canada, July 16-19, 2007, Proceedings, pages 65-74, 2007. Google Scholar
  11. Fedor V. Fomin, Pinar Heggernes, and Dieter Kratsch. Exact algorithms for graph homomorphisms. Theory Comput. Syst., 41(2):381-393, 2007. Google Scholar
  12. Martin Charles Golumbic and Udi Rotics. On the clique-width of some perfect graph classes. Int. J. Found. Comput. Sci., 11(3):423-443, 2000. Google Scholar
  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. P. Hell and Nešetřil. Graphs and homomorphisms, volume 28 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2004. Google Scholar
  15. P. Hell and J. Nešetřil. On the complexity of H-coloring. Journal of Combinatorial Theory, Ser.B, 48:92-110, 1990. Google Scholar
  16. Russell Impagliazzo and Ramamohan Paturi. Complexity of k-SAT. In Proceedings of the 14th Annual IEEE Conference on Computational Complexity, Atlanta, Georgia, USA, May 4-6, 1999, pages 237-240, 1999. Google Scholar
  17. Mikko Koivisto. An O^*(2ⁿ) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 583-590, 2006. Google Scholar
  18. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, 105:41-72, 2011. Google Scholar
  19. László Lovász. Kneser’s conjecture, chromatic number, and homotopy. J. Comb. Theory, Ser. A, 25(3):319-324, 1978. Google Scholar
  20. Johann A. Makowsky and Udi Rotics. On the clique-width of graphs with few P4’s. International Journal of Foundations of Computer Science, 10(03):329-348, 1999. Google Scholar
  21. Jesper Nederlof. Inclusion exclusion for hard problems. Master’s thesis, Department of Information and Computer Science, Utrecht University, 2008. URL: http://www.win.tue.nl/jnederlo/MScThesis.pdf.
  22. Patrick Traxler. The time complexity of constraint satisfaction. In Parameterized and Exact Computation, Third International Workshop, IWPEC 2008, Victoria, Canada, May 14-16, 2008. Proceedings, pages 190-201, 2008. Google Scholar
  23. Magnus Wahlström. New plain-exponential time classes for graph homomorphism. In Computer Science - Theory and Applications, Fourth International Computer Science Symposium in Russia, CSR 2009, Novosibirsk, Russia, August 18-23, 2009. Proceedings, pages 346-355, 2009. Google Scholar
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