Document Open Access Logo

Fine-Grained Complexity of the List Homomorphism Problem: Feedback Vertex Set and Cutwidth

Authors Marta Piecyk, Paweł Rzążewski

PDF
Thumbnail PDF

File

LIPIcs.STACS.2021.56.pdf
  • Filesize: 0.85 MB
  • 17 pages

Document Identifiers

Author Details

Marta Piecyk
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
Paweł Rzążewski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
  • Institute of Informatics, University of Warsaw, Poland

Funding

Supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.

Acknowledgements

We are grateful to Bart M. P. Jansen for introducing us to the problem and to Karolina Okrasa for many fruitful discussions.

Cite As Get BibTex

Marta Piecyk and Paweł Rzążewski. Fine-Grained Complexity of the List Homomorphism Problem: Feedback Vertex Set and Cutwidth. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 56:1-56:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.STACS.2021.56

Abstract

For graphs G,H, a homomorphism from G to H is an edge-preserving mapping from V(G) to V(H). In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is equipped with a list L(v) ⊆ V(H), and we need to determine whether there exists a homomorphism from G to H which additionally respects the lists L. List homomorphisms are a natural generalization of (list) colorings.
Very recently Okrasa, Piecyk, and Rzążewski [ESA 2020] studied the fine-grained complexity of the problem, parameterized by the treewidth of the instance graph G. They defined a new invariant i^*(H), and proved that for every relevant graph H, i.e., such that LHom(H) is NP-hard, this invariant is the correct base of the exponent in the running time of any algorithm solving the LHom(H) problem.
In this paper we continue this direction and study the complexity of the problem under different parameterizations. As the first result, we show that i^*(H) is also the right complexity base if the parameter is the size of a minimum feedback vertex set of G, denoted by fvs(G). In particular, for every relevant graph H, the LHom(H) problem  
- can be solved in time i^*(H)^fvs(G) ⋅ |V(G)|^𝒪(1), if a minimum feedback vertex set of G is given, 
- cannot be solved in time (i^*(H) - ε)^fvs(G) ⋅ |V(G)|^𝒪(1), for any ε > 0, unless the SETH fails.  Then we turn our attention to a parameterization by the cutwidth ctw(G) of G. Jansen and Nederlof [TCS 2019] showed that List k-Coloring (i.e., LHom(K_k)) can be solved in time c^ctw(G) ⋅ |V(G)|^𝒪(1) for an absolute constant c, i.e., the base of the exponential function does not depend on the number of colors. Jansen asked whether this behavior extends to graph homomorphisms. As the main result of the paper, we answer the question in the negative. We define a new graph invariant mim^*(H), closely related to the size of a maximum induced matching in H, and prove that for all relevant graphs H, the LHom(H) problem cannot be solved in time (mim^*(H)-ε)^{ctw(G)}⋅ |V(G)|^𝒪(1) for any ε > 0, unless the SETH fails. In particular, this implies that, assuming the SETH, there is no constant c, such that for every odd cycle the non-list version of the problem can be solved in time c^ctw(G) ⋅ |V(G)|^𝒪(1).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • list homomorphisms
  • fine-grained complexity
  • SETH
  • feedback vertex set
  • cutwidth

Metrics

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

Related Versions

References

  1. Noga Alon and Joel H. Spencer. The Probabilistic Method, Third Edition. Wiley-Interscience series in discrete mathematics and optimization. Wiley, 2008. Google Scholar
  2. Stefan Arnborg and Andrzej Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discret. Appl. Math., 23(1):11-24, 1989. URL: https://doi.org/10.1016/0166-218X(89)90031-0.
  3. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009. URL: https://doi.org/10.1137/070683933.
  4. H. L. Bodlaender. Classes of graphs with bounded tree-width. Bulletin of EATCS, pages 116-128, 1988. Google Scholar
  5. Hans L. Bodlaender and Arie M. C. A. Koster. Combinatorial optimization on graphs of bounded treewidth. Comput. J., 51(3):255-269, May 2008. URL: https://doi.org/10.1093/comjnl/bxm037.
  6. Fan R. K. Chung and Paul D. Seymour. Graphs with small bandwidth and cutwidth. Discret. Math., 75(1-3):113-119, 1989. URL: https://doi.org/10.1016/0012-365X(89)90083-6.
  7. Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comput., 9(3):251-280, 1990. URL: https://doi.org/10.1016/S0747-7171(08)80013-2.
  8. Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub Pachocki, and Arkadiusz Socala. Tight lower bounds on graph embedding problems. J. ACM, 64(3):18:1-18:22, 2017. URL: https://doi.org/10.1145/3051094.
  9. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  10. László Egri, Dániel Marx, and Paweł Rzążewski. Finding list homomorphisms from bounded-treewidth graphs to reflexive graphs: a complete complexity characterization. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, volume 96 of LIPIcs, pages 27:1-27:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.27.
  11. Tomás Feder and Pavol Hell. List homomorphisms to reflexive graphs. Journal of Combinatorial Theory, Series B, 72(2):236-250, 1998. URL: https://doi.org/10.1006/jctb.1997.1812.
  12. Tomás Feder, Pavol Hell, and Jing Huang. List homomorphisms and circular arc graphs. Combinatorica, 19(4):487-505, 1999. URL: https://doi.org/10.1007/s004939970003.
  13. Tomás Feder, Pavol Hell, and Jing Huang. Bi-arc graphs and the complexity of list homomorphisms. Journal of Graph Theory, 42(1):61-80, 2003. URL: https://doi.org/10.1002/jgt.10073.
  14. 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.
  15. M.R. Garey, D.S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976. URL: https://doi.org/10.1016/0304-3975(76)90059-1.
  16. Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Cliquewidth III: the odd case of graph coloring parameterized by cliquewidth. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 262-273. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.19.
  17. Pavol Hell and Jaroslav Nešetřil. Graphs and homomorphisms. Oxford University Press, 2004. Google Scholar
  18. 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.
  19. Ian Holyer. The NP-completeness of edge-coloring. SIAM J. Comput., 10(4):718-720, 1981. URL: https://doi.org/10.1137/0210055.
  20. Shenwei Huang. Improved complexity results on k-coloring P_t-free graphs. Eur. J. Comb., 51:336-346, 2016. URL: https://doi.org/10.1016/j.ejc.2015.06.005.
  21. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  22. 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.
  23. Lars Jaffke and Bart M. P. Jansen. Fine-grained parameterized complexity analysis of graph coloring problems. In Dimitris Fotakis, Aris Pagourtzis, and Vangelis Th. Paschos, editors, Algorithms and Complexity - 10th International Conference, CIAC 2017, Athens, Greece, May 24-26, 2017, Proceedings, volume 10236 of Lecture Notes in Computer Science, pages 345-356, 2017. URL: https://doi.org/10.1007/978-3-319-57586-5_29.
  24. Bart M. P. Jansen. Personal communication. Google Scholar
  25. Bart M. P. Jansen and Jesper Nederlof. Computing the chromatic number using graph decompositions via matrix rank. Theor. Comput. Sci., 795:520-539, 2019. URL: https://doi.org/10.1016/j.tcs.2019.08.006.
  26. Sanjeev Khanna, Nathan Linial, and Shmuel Safra. On the hardness of approximating the chromatic number. Combinatorica, 20(3):393-415, 2000. URL: https://doi.org/10.1007/s004930070013.
  27. Michael Lampis. Finer tight bounds for coloring on clique-width. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 86:1-86:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.86.
  28. Benoît Larose. Families of strongly projective graphs. Discuss. Math. Graph Theory, 22(2):271-292, 2002. URL: https://doi.org/10.7151/dmgt.1175.
  29. Benoit Larose and Claude Tardif. Strongly rigid graphs and projectivity. Multiple-Valued Logic, 7:339-361, 2001. Google Scholar
  30. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known algorithms on graphs of bounded treewidth are probably optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018. URL: https://doi.org/10.1145/3170442.
  31. Tomasz Łuczak and Jaroslav Nešetřil. Note on projective graphs. Journal of Graph Theory, 47(2):81-86, 2004. Google Scholar
  32. 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.
  33. Karolina Okrasa, Marta Piecyk, and Paweł Rzążewski. Full complexity classification of the list homomorphism problem for bounded-treewidth graphs. CoRR, abs/2006.11155, 2020. URL: http://arxiv.org/abs/2006.11155.
  34. Karolina Okrasa and Paweł Rzążewski. Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, pages 1578-1590, 2020. URL: https://doi.org/10.1137/1.9781611975994.97.
  35. Marta Piecyk and Paweł Rzążewski. Fine-grained complexity of the list homomorphism problem: feedback vertex set and cutwidth. CoRR, abs/2009.11642, 2020. URL: http://arxiv.org/abs/2009.11642.
  36. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  37. Paweł Rzążewski. Exact algorithm for graph homomorphism and locally injective graph homomorphism. Inf. Process. Lett., 114(7):387-391, 2014. URL: https://doi.org/10.1016/j.ipl.2014.02.012.
  38. 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.
  39. Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 887-898. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214056.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact