Document Open Access Logo

QCSP on Reflexive Tournaments

Authors Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, Stanislav Živný

PDF
Thumbnail PDF

File

LIPIcs.ESA.2021.58.pdf
  • Filesize: 0.72 MB
  • 15 pages

Document Identifiers

Author Details

Benoît Larose
  • LACIM, University of Québec, Montréal, Canada
Petar Marković
  • Department of Mathematics and Informatics, University of Novi Sad, Serbia
Barnaby Martin
  • Department of Computer Science, Durham University, UK
Daniël Paulusma
  • Department of Computer Science, Durham University, UK
Siani Smith
  • Department of Computer Science, Durham University, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

  • Živný, Stanislav: Stanislav Zivny was supported by a Royal Society University Research Fellowship. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 714532). The paper reflects only the authors’ views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.

Acknowledgements

We are grateful to several referees for careful reading of the paper and good advice.

Cite AsGet BibTex

Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.58

Abstract

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • computational complexity
  • algorithmic graph theory
  • quantified constraints
  • universal algebra
  • constraint satisfaction

Metrics

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

Related Versions

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. Ferdinand Börner, Andrei A. Bulatov, Hubie Chen, Peter Jeavons, and Andrei A. Krokhin. The complexity of constraint satisfaction games and QCSP. Inf. Comput., 207(9):923-944, 2009. URL: https://doi.org/10.1016/j.ic.2009.05.003.
  4. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 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. Comptes Rendus de l'Académie des Sciences Paris, 249:2151-2152, 1959. Google Scholar
  7. Hubie Chen. The complexity of quantified constraint satisfaction: Collapsibility, sink algebras, and the three-element case. SIAM J. Comput., 37(5):1674-1701, 2008. URL: https://doi.org/10.1137/060668572.
  8. Hubie Chen, Florent R. Madelaine, and Barnaby Martin. Quantified constraints and containment problems. Logical Methods in Computer Science, 11(3), 2015. Extended abstract appeared at LICS 2008. This journal version incorporates principal part of CP 2012 Containment, Equivalence and Coreness from CSP to QCSP and Beyond. URL: https://doi.org/10.2168/LMCS-11(3:9)2015.
  9. Víctor Dalmau and Andrei A. Krokhin. Majority constraints have bounded pathwidth duality. European Journal of Combinatorics, 29(4):821-837, 2008. Google Scholar
  10. Petar Dapic, Petar Markovic, and Barnaby Martin. Quantified constraint satisfaction problem on semicomplete digraphs. ACM Trans. Comput. Log., 18(1):2:1-2:47, 2017. URL: https://doi.org/10.1145/3007899.
  11. 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
  12. 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
  13. P. G. Kolaitis and M. Y. Vardi. Finite Model Theory and Its Applications (Texts in Theoretical Computer Science. An EATCS Series), chapter A logical Approach to Constraint Satisfaction, pages 339-370. Springer-Verlag New York, Inc., 2005. Google Scholar
  14. Benoit Larose. Taylor operations on finite reflexive structures. International Journal of Mathematics and Computer Science, 1(1):1-26, 2006. Google Scholar
  15. Benoit Larose, Barnaby Martin, and Daniël Paulusma. Surjective H-colouring over reflexive digraphs. TOCT, 11(1):3:1-3:21, 2019. URL: https://doi.org/10.1145/3282431.
  16. Benoit Larose and László Zádori. Finite posets and topological spaces in locally finite varieties. Algebra Universalis, 52(2):119-136, 2005. Google Scholar
  17. Barnaby Martin. QCSP on partially reflexive forests. In Principles and Practice of Constraint Programming - CP 2011 - 17th International Conference, CP 2011, Perugia, Italy, September 12-16, 2011. Proceedings, pages 546-560, 2011. URL: https://doi.org/10.1007/978-3-642-23786-7_42.
  18. Barnaby Martin and Florent R. Madelaine. Towards a trichotomy for quantified H-coloring. In Logical Approaches to Computational Barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006, Proceedings, pages 342-352, 2006. URL: https://doi.org/10.1007/11780342_36.
  19. Narayan Vikas. Algorithms for partition of some class of graphs under compaction and vertex-compaction. Algorithmica, 67(2):180-206, 2013. Google Scholar
  20. Narayan Vikas. Computational complexity of graph partition under vertex-compaction to an irreflexive hexagon. In 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, pages 69:1-69:14, 2017. Google Scholar
  21. Alexander Wires. Dichotomy for finite tournaments of mixed-type. Discrete Mathematics, 338(12):2523-2538, 2015. URL: https://doi.org/10.1016/j.disc.2015.06.024.
  22. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342, 2017. Google Scholar
  23. Dmitriy Zhuk. No-rainbow problem and the surjective constraint satisfaction problem, 2020. To appear LICS 2021. URL: http://arxiv.org/abs/2003.11764.
  24. Dmitriy Zhuk. A proof of the CSP dichotomy conjecture. J. ACM, 67(5):30:1-30:78, 2020. URL: https://doi.org/10.1145/3402029.
  25. Dmitriy Zhuk and Barnaby Martin. QCSP monsters and the demise of the Chen Conjecture. In Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, and Julia Chuzhoy, editors, Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 91-104. ACM, 2020. URL: https://doi.org/10.1145/3357713.3384232.
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