Document Open Access Logo

Surjective H-Colouring over Reflexive Digraphs

Authors Benoit Larose, Barnaby Martin, Daniel Paulusma

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2018.49.pdf
  • Filesize: 0.51 MB
  • 14 pages

Document Identifiers

Subject Classification

Keywords
  • Surjective H-Coloring
  • Computational Complexity
  • Algorithmic Graph Theory
  • Universal Algebra
  • Constraint Satisfaction

Metrics

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

Abstract

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. 

Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.

By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

Cite As Get BibTex

Benoit Larose, Barnaby Martin, and Daniel Paulusma. Surjective H-Colouring over Reflexive Digraphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.STACS.2018.49

Author Details

Benoit Larose
Barnaby Martin
Daniel Paulusma

References

  1. Jørgen Bang-Jensen, Pavol Hell, and Gary MacGillivray. The complexity of colouring by semicomplete digraphs. SIAM Journal on Discrete Mathematics, 1(3):281-298, 1988. Google Scholar
  2. Manuel Bodirsky, Jan Kára, and Barnaby Martin. The complexity of surjective homomorphism problems - a survey. Discrete Applied Mathematics, 160(12):1680-1690, 2012. Google Scholar
  3. Andrei A. Bulatov. Complexity of conservative constraint satisfaction problems. ACM Transactions on Computational Logic, 12(4):24:1-24:66, 2011. Google Scholar
  4. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. Proc. FOCS 2017, pages 319-330, 2017. Google Scholar
  5. Andrei A. Bulatov, Peter Jeavons, and Andrei A. Krokhin. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34(3):720-742, 2005. Google Scholar
  6. Paul Camion. Chemins et circuits hamiltoniens de graphes complets. C. R. Acad. Sci. Paris, 249:2151-2152, 1959. Google Scholar
  7. Hubie Chen. An algebraic hardness criterion for surjective constraint satisfaction. Algebra Universalis, 72(4):393-401, 2014. Google Scholar
  8. Víctor Dalmau and Andrei A. Krokhin. Majority constraints have bounded pathwidth duality. European Journal of Combinatorics, 29(4):821-837, 2008. Google Scholar
  9. Tomás Feder and Pavol Hell. List homomorphisms to reflexive graphs. Journal of Combinatorial Theory, Series B, 72(2):236-250, 1998. Google Scholar
  10. Tomás Feder, Pavol Hell, and Jing Huang. List homomorphisms and circular arc graphs. Combinatorica, 19(4):487-505, 1999. Google Scholar
  11. 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. Google Scholar
  12. Tomás Feder, Pavol Hell, Peter Jonsson, Andrei A. Krokhin, and Gustav Nordh. Retractions to pseudoforests. SIAM Journal on Discrete Mathematics, 24(1):101-112, 2010. Google Scholar
  13. Tomás Feder, Pavol Hell, and Kim Tucker-Nally. Digraph matrix partitions and trigraph homomorphisms. Discrete Applied Mathematics, 154(17):2458-2469, 2006. Google Scholar
  14. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1998. Google Scholar
  15. Petr A. Golovach, Matthew Johnson, Barnaby Martin, Daniël Paulusma, and Anthony Stewart. Surjective H-colouring: New hardness results. Proc. CiE 2017, Lecture Notes in Computer Science, 10307:270-281, 2017. Google Scholar
  16. Petr A. Golovach, Daniël Paulusma, and Jian Song. Computing vertex-surjective homomorphisms to partially reflexive trees. Theoretical Computer Science, 457:86-100, 2012. Google Scholar
  17. David Charles Hobby and Ralph McKenzie. The Structure of Finite Algebras, Contemporary Mathematics. American Mathematical Society, 1988. Google Scholar
  18. Daniel Král, Jan Kratochvíl, Andrzej Proskurowski, and Heinz-Jürgen Voss. Coloring mixed hypertrees. Discrete Applied Mathematics, 154(4):660-672, 2006. Google Scholar
  19. Benoit Larose. Taylor operations on finite reflexive structures. International Journal of Mathematics and Computer Science, 1(1):1-26, 2006. Google Scholar
  20. Benoit Larose. Algebra and the complexity of digraph CSPs: a survey. In Andrei A. Krokhin and Stanislav Zivny, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 267-285. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  21. Benoit Larose, Cynthia Loten, and Claude Tardif. A characterisation of first-order constraint satisfaction problems. Logical Methods in Computer Science, 3(4), 2007. Google Scholar
  22. Benoit Larose and László Zádori. Finite posets and topological spaces in locally finite varieties. Algebra Universalis, 52(2):119-136, 2005. Google Scholar
  23. Miklós Maróti and László Zádori. Reflexive digraphs with near unanimity polymorphisms. Discrete Mathematics, 312(15):2316-2328, 2012. Google Scholar
  24. Barnaby Martin and Daniël Paulusma. The computational complexity of disconnected cut and 2_k2-partition. Journal of Combinatorial Theory, Series B, 111:17-37, 2015. Google Scholar
  25. Mark Siggers. personal communication. Google Scholar
  26. Agnes Szendrei. Term minimal algebras. Algebra Universalis, 32(4):439-477, 1994. Google Scholar
  27. Narayan Vikas. Computational complexity of compaction to reflexive cycles. SIAM Journal on Computing, 32(1):253-280, 2002. Google Scholar
  28. Narayan Vikas. Computational complexity of compaction to irreflexive cycles. Journal of Computer and System Sciences, 68(3):473-496, 2004. Google Scholar
  29. Narayan Vikas. A complete and equal computational complexity classification of compaction and retraction to all graphs with at most four vertices and some general results. Journal of Computer and System Sciences, 71(4):406-439, 2005. Google Scholar
  30. Narayan Vikas. Algorithms for partition of some class of graphs under compaction and vertex-compaction. Algorithmica, 67(2):180-206, 2013. Google Scholar
  31. Narayan Vikas. Computational complexity of graph partition under vertex-compaction to an irreflexive hexagon. In Kim G. Larsen, Hans L. Bodlaender, and Jean-François Raskin, editors, 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, volume 83 of LIPIcs, pages 69:1-69:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. Google Scholar
  32. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. Proc. FOCS 2017, pages 331-342, 2017. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact