Document Open Access Logo

Sparsification Lower Bounds for List H-Coloring

Authors Hubie Chen, Bart M. P. Jansen , Karolina Okrasa , Astrid Pieterse , Paweł Rzążewski

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2020.58.pdf
  • Filesize: 0.76 MB
  • 17 pages

Document Identifiers

Author Details

Hubie Chen
  • Birkbeck, University of London, Malet Street, Bloomsbury, UK
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Karolina Okrasa
  • University of Warsaw, Institute of Informatics, Poland
  • Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland
Astrid Pieterse
  • Department of Computer Science, Humboldt-Universität zu Berlin, Germany
Paweł Rzążewski
  • Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland
  • University of Warsaw, Institute of Informatics, Poland

Funding

  • Jansen, Bart M. P.: Supported by NWO Gravitation grant "Networks".
  • Okrasa, Karolina: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreement no. 714704.
  • Pieterse, Astrid: Supported by DFG Emmy Noether-grant (KR 4286/1), part of this research was supported by NWO Gravitation grant "Networks".
  • Rzążewski, Paweł: Supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.

Acknowledgements

We acknowledge the productive atmosphere at Dagstuhl Seminar 19271 "Graph Colouring: from Structure to Algorithms", where this work has been initiated.

Cite AsGet BibTex

Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, and Paweł Rzążewski. Sparsification Lower Bounds for List H-Coloring. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 58:1-58:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.58

Abstract

We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V(G) is mapped to a vertex on its list L(v) ⊆ V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize 𝒪(n^(2-ε)) for some ε > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless NP ⊆ coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-arc graphs.

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 H-Coloring
  • Sparsification
  • Constraint Satisfaction Problem

Metrics

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

Related Versions

References

  1. Michael O. Albertson, Paul A. Catlin, and Luana Gibbons. Homomorphisms of 3-chromatic graphs. II. In Proceedings of the 16th Southeastern international conference on combinatorics, graph theory and computing, volume 47, pages 19-28, 1985. Google Scholar
  2. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014. URL: https://doi.org/10.1137/120880240.
  3. Andrei A. Bulatov. H-coloring dichotomy revisited. Theor. Comput. Sci., 349(1):31-39, 2005. URL: https://doi.org/10.1016/j.tcs.2005.09.028.
  4. Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, and Paweł Rzążewski. Sparsification lower bounds for List H-Coloring. CoRR, abs/2009.08353, 2020. URL: http://arxiv.org/abs/2009.08353.
  5. Hubie Chen, Bart M. P. Jansen, and Astrid Pieterse. Best-case and worst-case sparsifiability of boolean CSPs. In Proceedings of the 13th International Symposium on Parameterized and Exact Computation, volume 115 of LIPIcs, pages 15:1-15:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.IPEC.2018.15.
  6. Víctor Dalmau, László Egri, Pavol Hell, Benoît Larose, and Arash Rafiey. Descriptive complexity of List H-Coloring problems in logspace: A refined dichotomy. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015, pages 487-498. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/LICS.2015.52.
  7. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: https://doi.org/10.1145/2629620.
  8. László Egri, Pavol Hell, Benoît Larose, and Arash Rafiey. Space complexity of list H-colouring: a dichotomy. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 349-365. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.26.
  9. László Egri, Andrei A. Krokhin, Benoît Larose, and Pascal Tesson. The complexity of the list homomorphism problem for graphs. Theory Comput. Syst., 51(2):143-178, 2012. URL: https://doi.org/10.1007/s00224-011-9333-8.
  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. Louis Esperet, Mickaël Montassier, Pascal Ochem, and Alexandre Pinlou. A complexity dichotomy for the coloring of sparse graphs. J. Graph Theory, 73(1):85-102, 2013. URL: https://doi.org/10.1002/jgt.21659.
  12. Tomás Feder and Pavol Hell. List homomorphisms to reflexive graphs. J. Comb. Theory, Ser. 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):386-392, 2007. Algebraic and Topological Methods in Graph Theory. URL: https://doi.org/10.1016/j.disc.2005.09.030.
  16. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, December 2018. URL: https://doi.org/10.1017/9781107415157.
  17. Anna Galluccio, Pavol Hell, and Jaroslav Nešetřil. The complexity of H-colouring of bounded degree graphs. Discrete Math., 222(1-3):101-109, 2000. URL: https://doi.org/10.1016/S0012-365X(00)00009-1.
  18. Carla Groenland, Karolina Okrasa, Paweł Rzążewski, Alex D. Scott, Paul D. 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.
  19. Jiong Guo and Rolf Niedermeier. Invitation to data reduction and problem kernelization. SIGACT News, 38(1):31-45, 2007. URL: https://doi.org/10.1145/1233481.1233493.
  20. Geňa Hahn and Claude Tardif. Graph homomorphisms: structure and symmetry, pages 107-166. Springer Netherlands, 1997. URL: https://doi.org/10.1007/978-94-015-8937-6_4.
  21. 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.
  22. Pavol Hell and Jaroslav Nešetřil. Graphs and Homomorphisms. Oxford Lecture Series in Mathematics. Oxford University Press, 2004. Google Scholar
  23. Robert W. Irving. NP-completeness of a family of graph-colouring problems. Discrete Appl. Math., 5(1):111-117, 1983. URL: https://doi.org/10.1016/0166-218X(83)90020-3.
  24. Lars Jaffke and Bart M. P. Jansen. Fine-grained parameterized complexity analysis of graph coloring problems. In Proceedings of the 10th International Conference on Algorithms and Complexity, pages 345-356, 2017. URL: https://doi.org/10.1007/978-3-319-57586-5_29.
  25. Bart M. P. Jansen. On sparsification for computing treewidth. Algorithmica, 71(3):605-635, 2015. URL: https://doi.org/10.1007/s00453-014-9924-2.
  26. Bart M. P. Jansen and Astrid Pieterse. Optimal data reduction for graph coloring using low-degree polynomials. In 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, September 6-8, 2017, Vienna, Austria, volume 89 of LIPIcs, pages 22:1-22:12, 2017. URL: https://doi.org/10.4230/LIPIcs.IPEC.2017.22.
  27. Bart M. P. Jansen and Astrid Pieterse. Sparsification upper and lower bounds for graph problems and not-all-equal SAT. Algorithmica, 79(1):3-28, 2017. URL: https://doi.org/10.1007/s00453-016-0189-9.
  28. Bart M. P. Jansen and Astrid Pieterse. Optimal data reduction for graph coloring using low-degree polynomials. Algorithmica, 81(10):3865-3889, 2019. URL: https://doi.org/10.1007/s00453-019-00578-5.
  29. Gábor Kun and Mario Szegedy. A new line of attack on the dichotomy conjecture. Eur. J. Comb., 52:338-367, 2016. URL: https://doi.org/10.1016/j.ejc.2015.07.011.
  30. Victor Lagerkvist and Magnus Wahlström. Kernelization of constraint satisfaction problems: A study through universal algebra. In Proceedings of the 23rd International Conference on Principles and Practice of Constraint Programming, volume 10416, pages 157-171. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-66158-2_11.
  31. Gary MacGillivray and Mark H. Siggers. On the complexity of H-colouring planar graphs. Discrete Math., 309(18):5729-5738, 2009. URL: https://doi.org/10.1016/j.disc.2008.05.016.
  32. H.A. Maurer, J.H. Sudborough, and E. Welzl. On the complexity of the general coloring problem. Information and Control, 51(2):128-145, 1981. URL: https://doi.org/10.1016/S0019-9958(81)90226-6.
  33. Jaroslav Nešetřil. Representations of graphs by means of products and their complexity. In Proceedings of the 10th International Symposium on Mathematical Foundations of Computer Science, volume 118 of Lecture Notes in Computer Science, pages 94-102, 1981. URL: https://doi.org/10.1007/3-540-10856-4_76.
  34. 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.
  35. 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.
  36. Karolina Okrasa and Paweł Rzążewski. Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1578-1590. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.97.
  37. Karolina Okrasa and Paweł Rzążewski. Subexponential algorithms for variants of the homomorphism problem in string graphs. J. Comput. Syst. Sci., 109:126-144, 2020. URL: https://doi.org/10.1016/j.jcss.2019.12.004.
  38. Astrid Pieterse. Tight parameterized preprocessing bounds: sparsification via low-degree polynomials. PhD thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, September 2019. Proefschrift. Google Scholar
  39. M. Siggers. A new proof of the H-coloring dichotomy. SIAM J. Discrete Math., 23(4):2204-2210, 2010. Corrected manuscript available from the author’s homepage. URL: https://doi.org/10.1137/080736697.
  40. Mark H. Siggers. Dichotomy for bounded degree H-colouring. Discrete Appl. Math., 157(2):201-210, 2009. URL: https://doi.org/10.1016/j.dam.2008.02.003.
  41. William T. Trotter and John I. Moore. Characterization problems for graphs, partially ordered sets, lattices, and families of sets. Discrete Mathematics, 16(4):361-381, 1976. URL: https://doi.org/10.1016/S0012-365X(76)80011-8.
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 Imprint Privacy Contact