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List Locally Surjective Homomorphisms in Hereditary Graph Classes

Authors Pavel Dvořák , Tomáš Masařík , Jana Novotná , Monika Krawczyk, Paweł Rzążewski , Aneta Żuk



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Pavel Dvořák
  • Tata Institute of Fundamental Research, Mumbai, India
  • Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Tomáš Masařík
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Jana Novotná
  • Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Monika Krawczyk
  • 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, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Aneta Żuk
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland

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Pavel Dvořák, Tomáš Masařík, Jana Novotná, Monika Krawczyk, Paweł Rzążewski, and Aneta Żuk. List Locally Surjective Homomorphisms in Hereditary Graph Classes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 30:1-30:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.30

Abstract

A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time. We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family 𝒮 consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P₃,C₄} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ∈ 𝒮 (unless the ETH fails).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Homomorphism
  • Hereditary graphs
  • Subexponential-time algorithms

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References

  1. Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Oliver Schaudt, Maya Stein, and Mingxian Zhong. Three-coloring and list three-coloring of graphs without induced paths on seven vertices. Combinatorica, 38(4):779-801, 2018. URL: https://doi.org/10.1007/s00493-017-3553-8.
  2. Laurent Bulteau, Konrad K. Dabrowski, Noleen Köhler, Sebastian Ordyniak, and Daniël Paulusma. An algorithmic framework for locally constrained homomorphisms. CoRR, abs/2201.11731, 2022. URL: https://doi.org/10.48550/arXiv.2201.11731.
  3. Eglantine Camby and Oliver Schaudt. A new characterization of P_k-free graphs. Algorithmica, 75(1):205-217, 2016. URL: https://doi.org/10.1007/s00453-015-9989-6.
  4. Steven Chaplick, Jiří Fiala, Pim van 't Hof, Daniël Paulusma, and Marek Tesař. Locally constrained homomorphisms on graphs of bounded treewidth and bounded degree. Theoretical Computer Science, 590:86-95, 2015. URL: https://doi.org/10.1016/j.tcs.2015.01.028.
  5. Maria Chudnovsky, Shenwei Huang, Pawel Rzążewski, Sophie Spirkl, and Mingxian Zhong. Complexity of C_k-coloring in hereditary classes of graphs. In Michael A. Bender, Ola Svensson, and Grzegorz Herman, editors, 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, volume 144 of LIPIcs, pages 31:1-31:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.31.
  6. Maria Chudnovsky, Shenwei Huang, Sophie Spirkl, and Mingxian Zhong. List 3-coloring graphs with no induced P₆+P₃. Algorithmica, 83(1):216-251, 2021. URL: https://doi.org/10.1007/s00453-020-00754-y.
  7. Maria Chudnovsky, Jason King, Michał Pilipczuk, Paweł Rzążewski, and Sophie Spirkl. Finding large h-colorable subgraphs in hereditary graph classes. SIAM Journal on Discrete Mathematics, 35(4):2357-2386, 2021. URL: https://doi.org/10.1137/20M1367660.
  8. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems, 33(2):125-150, 2000. URL: https://doi.org/10.1007/s002249910009.
  9. Pavel Dvořák, Monika Krawczyk, Tomáš Masařík, Jana Novotná, Paweł Rzążewski, and Aneta Żuk. List locally surjective homomorphisms in hereditary graph classes, 2022. URL: https://doi.org/10.48550/arXiv.2202.12438.
  10. Thomas Emden-Weinert, Stefan Hougardy, and Bernd Kreuter. Uniquely colourable graphs and the hardness of colouring graphs of large girth. Combinatorics, Probability and Computing, 7(4):375-386, 1998. URL: https://doi.org/10.1017/S0963548398003678.
  11. Martin G. Everett and Steve Borgatti. Role colouring a graph. Mathematical Social Sciences, 21(2):183-188, 1991. URL: https://doi.org/10.1016/0165-4896(91)90080-B.
  12. 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.
  13. 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.
  14. 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.
  15. Tomás Feder, Pavol Hell, and Jing Huang. List homomorphisms of graphs with bounded degrees. Discrete Mathematics, 307(3-5):386-392, 2007. URL: https://doi.org/10.1016/j.disc.2005.09.030.
  16. Jiří Fiala and Jan Kratochvíl. Locally constrained graph homomorphisms - structure, complexity, and applications. Computer Science Review, 2(2):97-111, 2008. URL: https://doi.org/10.1016/j.cosrev.2008.06.001.
  17. Jiří Fiala and Daniël Paulusma. A complete complexity classification of the role assignment problem. Journal of Computer and System Sciences, 349(1):67-81, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.029.
  18. Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A survey on the computational complexity of coloring graphs with forbidden subgraphs. Journal of Graph Theory, 84(4):331-363, 2017. URL: https://doi.org/10.1002/jgt.22028.
  19. Petr A. Golovach, Daniël Paulusma, and Jian Song. Closing complexity gaps for coloring problems on H-free graphs. Information and Computation, 237:204-214, 2014. URL: https://doi.org/10.1016/j.ic.2014.02.004.
  20. Carla Groenland, Karolina Okrasa, Paweł Rzążewski, Alex Scott, Paul Seymour, and Sophie Spirkl. H-colouring P_t-free graphs in subexponential time. Discrete Applied Mathematics, 267:184-189, 2019. URL: https://doi.org/10.1016/j.dam.2019.04.010.
  21. Pavol Hell and Jaroslav Nešetřil. On the complexity of H-coloring. Journal of Combinatorial Theory, Series B, 48(1):92-110, 1990. URL: https://doi.org/10.1016/0095-8956(90)90132-J.
  22. Chính T. Hoàng, Marcin Kaminski, Vadim V. Lozin, Joe Sawada, and Xiao Shu. Deciding k-colorability of P₅-free graphs in polynomial time. Algorithmica, 57(1):74-81, 2010. URL: https://doi.org/10.1007/s00453-008-9197-8.
  23. Ian Holyer. The NP-completeness of edge-coloring. SIAM Journal on Computing, 10(4):718-720, 1981. URL: https://doi.org/10.1137/0210055.
  24. Shenwei Huang. Improved complexity results on k-coloring P_t-free graphs. European Journal of Combinatorics, 51:336-346, 2016. URL: https://doi.org/10.1016/j.ejc.2015.06.005.
  25. 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.
  26. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  27. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  28. Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná, Daniël Paulusma, and Veronika Slívová. Colouring P_r+P_s-free graphs. Algorithmica, 82(7):1833-1858, 2020. URL: https://doi.org/10.1007/s00453-020-00675-w.
  29. Daniel Leven and Zvi Galil. NP-completeness of finding the chromatic index of regular graphs. Journal of Algorithms, 4(1):35-44, 1983. URL: https://doi.org/10.1016/0196-6774(83)90032-9.
  30. László Lovász. Coverings and colorings of hypergraphs. In Proc. 4th Southeastern Conference of Combinatorics, Graph Theory, and Computing, volume 37 of Utilitas Math., pages 3-12, 1973. Google Scholar
  31. Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, Erik Jan van Leeuwen, and Bartosz Walczak. Subexponential-time algorithms for finding large induced sparse subgraphs. Algorithmica, 83(8):2634-2650, 2021. URL: https://doi.org/10.1007/s00453-020-00745-z.
  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 and Paweł Rzążewski. Complexity of the list homomorphism problem in hereditary graph classes. CoRR, abs/2010.03393, 2020. URL: http://arxiv.org/abs/2010.03393.
  34. Karolina Okrasa and Paweł Rzążewski. Subexponential algorithms for variants of the homomorphism problem in string graphs. Journal of Computer and System Sciences, 109:126-144, 2020. URL: https://doi.org/10.1016/j.jcss.2019.12.004.
  35. Karolina Okrasa and Paweł Rzążewski. Complexity of the list homomorphism problem in hereditary graph classes. 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 54:1-54:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.54.
  36. 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. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.56.
  37. Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Quasi-polynomial-time algorithm for Independent Set in P_t-free graphs via shrinking the space of induced paths. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 204-209. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.23.
  38. Sophie Spirkl, Maria Chudnovsky, and Mingxian Zhong. Four-coloring P₆-free graphs. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1239-1256. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.76.
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