An Order out of Nowhere: A New Algorithm for Infinite-Domain {CSP}s

Authors Antoine Mottet , Tomáš Nagy , Michael Pinsker



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.148.pdf
  • Filesize: 0.8 MB
  • 18 pages

Document Identifiers

Author Details

Antoine Mottet
  • Research Group on Theoretical Computer Science, Hamburg University of Technology, Germany
Tomáš Nagy
  • Theoretical Computer Science Department, Jagiellonian University, Kraków, Poland
Michael Pinsker
  • Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Austria

Cite As Get BibTex

Antoine Mottet, Tomáš Nagy, and Michael Pinsker. An Order out of Nowhere: A New Algorithm for Infinite-Domain {CSP}s. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 148:1-148:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.148

Abstract

We consider the problem of satisfiability of sets of constraints in a given set of finite uniform hypergraphs. While the problem under consideration is similar in nature to the problem of satisfiability of constraints in graphs, the classical complexity reduction to finite-domain CSPs that was used in the proof of the complexity dichotomy for such problems cannot be used as a black box in our case. We therefore introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration. Our second main result is a P/NP-complete complexity dichotomy for such problems over many sets of uniform hypergraphs. The proof is based on the translation of the problem into the framework of constraint satisfaction problems (CSPs) over infinite uniform hypergraphs. Our result confirms in particular the Bodirsky-Pinsker conjecture for CSPs of first-order reducts of some homogeneous hypergraphs. This forms a vast generalization of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongrácz (ICALP'16) on graph satisfiability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic
Keywords
  • Constraint Satisfaction Problems
  • Hypergraphs
  • Polymorphisms

Metrics

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

References

  1. Reza Akhtar and Alistair H. Lachlan. On countable homogeneous 3-hypergraphs. Archive for Mathematical Logic, 34:331-344, 1995. URL: https://doi.org/10.1007/BF01387512.
  2. Libor Barto, Michael Kompatscher, Miroslav Olšák, Trung Van Pham, and Michael Pinsker. The equivalence of two dichotomy conjectures for infinite domain constraint satisfaction problems. In Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS'17, 2017. URL: https://doi.org/10.1109/LICS.2017.8005128.
  3. Libor Barto, Michael Kompatscher, Miroslav Olšák, Trung Van Pham, and Michael Pinsker. Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures. Journal of Mathematical Logic, 19(02):1950010, 2019. URL: https://doi.org/10.1142/S0219061319500107.
  4. Libor Barto and Marcin Kozik. Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Log. Methods Comput. Sci., 8(1), 2012. URL: https://doi.org/10.2168/LMCS-8(1:7)2012.
  5. Libor Barto, Jakub Opršal, and Michael Pinsker. The wonderland of reflections. Israel Journal of Mathematics, 223(1):363-398, 2018. URL: https://doi.org/10.1007/s11856-017-1621-9.
  6. Manuel Bodirsky, Peter Jonsson, and Trung Van Pham. The complexity of phylogeny constraint satisfaction problems. ACM Transactions on Computational Logic, 18(3), 2017. An extended abstract appeared in the conference STACS 2016. URL: https://doi.org/10.1145/3105907.
  7. Manuel Bodirsky and Jan Kára. The complexity of equality constraint languages. Theory of Computing Systems, 3(2):136-158, 2008. A conference version appeared in the proceedings of Computer Science Russia (CSR'06). URL: https://doi.org/10.1007/S00224-007-9083-9.
  8. Manuel Bodirsky and Jan Kára. The complexity of temporal constraint satisfaction problems. Journal of the ACM, 57(2), 2010. An extended abstract appeared in the Proceedings of the Symposium on Theory of Computing (STOC). URL: https://doi.org/10.1145/1667053.1667058.
  9. Manuel Bodirsky and Simon Knäuer. Network satisfaction for symmetric relation algebras with a flexible atom. Proceedings of the 35th AAAI Conference on Artificial Intelligence, 35(7):6218-6226, 2021. URL: https://doi.org/10.1609/AAAI.V35I7.16773.
  10. Manuel Bodirsky and Simon Knäuer. The complexity of network satisfaction problems for symmetric relation algebras with a flexible atom. J. Artif. Intell. Res., 75:1701-1744, 2022. URL: https://doi.org/10.1613/JAIR.1.14195.
  11. Manuel Bodirsky, Florent R. Madelaine, and Antoine Mottet. A universal-algebraic proof of the complexity dichotomy for monotone monadic SNP. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science - LICS'18, 2018. URL: https://doi.org/10.1145/3209108.3209156.
  12. Manuel Bodirsky, Barnaby Martin, Michael Pinsker, and András Pongrácz. Constraint satisfaction problems for reducts of homogeneous graphs. SIAM Journal on Computing, 48(4):1224-1264, 2019. A conference version appeared in the Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, pages 119:1-119:14. URL: https://doi.org/10.1137/16M1082974.
  13. Manuel Bodirsky and Antoine Mottet. Reducts of finitely bounded homogeneous structures, and lifting tractability from finite-domain constraint satisfaction. In Proceedings of the 31th Annual ACM/IEEE Symposium on Logic in Computer Science - LICS'16, 2016. URL: https://doi.org/10.1145/2933575.2934515.
  14. Manuel Bodirsky and Antoine Mottet. A dichotomy for first-order reducts of unary structures. Logical Methods in Computer Science, 14(2), 2018. URL: https://doi.org/10.23638/LMCS-14(2:13)2018.
  15. Manuel Bodirsky and Michael Pinsker. Schaefer’s theorem for graphs. Journal of the ACM, 62(3):19:1-19:52, 2015. A conference version appeared in the Proceedings of the Symposium on Theory of Computing (STOC 2011), pages 655-664. URL: https://doi.org/10.1145/2764899.
  16. Manuel Bodirsky and Michael Pinsker. Topological Birkhoff. Transactions of the American Mathematical Society, 367:2527-2549, 2015. URL: https://doi.org/10.1090/S0002-9947-2014-05975-8.
  17. Manuel Bodirsky, Michael Pinsker, and András Pongrácz. Projective clone homomorphisms. Journal of Symbolic Logic, 86(1):148-161, 2021. URL: https://doi.org/10.1017/jsl.2019.23.
  18. Bertalan Bodor. CSP dichotomy for ω-categorical monadically stable structures. PhD dissertation, Institute of Algebra, Technische Universität Dresden, 2021. URL: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-774379.
  19. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  20. Moses Charikar, Venkatesan Guruswami, and Rajsekar Manokaran. Every permutation CSP of arity 3 is approximation resistant. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15-18 July 2009, pages 62-73. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/CCC.2009.29.
  21. Vaggos Chatziafratis and Konstantin Makarychev. Phylogenetic CSPs are approximation resistant. CoRR, abs/2212.12765, 2022. URL: https://doi.org/10.48550/arXiv.2212.12765.
  22. Roland Fraïssé. Une hypothèse sur l'extension des relations finies et sa vérification dans certaines classes particulières (deuxième partie). Synthese, 16(1):34-46, 1966. URL: http://www.jstor.org/stable/20114493.
  23. 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. URL: https://doi.org/10.1137/S0097539794266766.
  24. Venkatesan Guruswami, Johan Håstad, Rajsekar Manokaran, Prasad Raghavendra, and Moses Charikar. Beating the random ordering is hard: Every ordering CSP is approximation resistant. SIAM J. Comput., 40(3):878-914, 2011. URL: https://doi.org/10.1137/090756144.
  25. Wilfrid Hodges. Model theory, volume 42 of Encyclopedia of mathematics and its applications. Cambridge University Press, 1993. Google Scholar
  26. Michael Kompatscher and Trung Van Pham. A complexity dichotomy for poset constraint satisfaction. IfCoLog Journal of Logics and their Applications (FLAP), 5(8):1663-1696, 2018. A conference version appeared in the Proceedings of the 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), pages 47:1-47:12. URL: https://www.collegepublications.co.uk/downloads/ifcolog00028.pdf.
  27. Alistair H. Lachlan and Robert E. Woodrow. Countable ultrahomogeneous undirected graphs. Transactions of the American Mathematical Society, 262, 1980. URL: https://doi.org/10.1090/S0002-9947-1980-0583847-2.
  28. Richard Emil Ladner. On the structure of polynomial time reducibility. J. ACM, 22(1):155-171, January 1975. URL: https://doi.org/10.1145/321864.321877.
  29. Antoine Mottet, Tomáš Nagy, and Michael Pinsker. An order out of nowhere: a new algorithm for infinite-domain csps, 2023. URL: https://arxiv.org/abs/2301.12977.
  30. Antoine Mottet and Michael Pinsker. Smooth approximations and CSPs over finitely bounded homogeneous structures. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science - LICS'22, 2022. URL: https://doi.org/10.1145/3531130.3533353.
  31. Antoine Mottet, Tomáš Nagy, Michael Pinsker, and Michał Wrona. Smooth approximations and relational width collapses. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 138:1-138:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.138.
  32. Antoine Mottet, Tomáš Nagy, Michael Pinsker, and Michał Wrona. When symmetries are enough: collapsing the bounded width hierarchy for infinite-domain CSPs. arxiv:2102.07531, 2022. URL: https://doi.org/10.48550/arXiv.2102.07531.
  33. George Osipov and Magnus Wahlström. Parameterized Complexity of Equality MinCSP. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, 31st Annual European Symposium on Algorithms, ESA 2023, September 4-6, 2023, Amsterdam, The Netherlands, volume 274 of LIPIcs, pages 86:1-86:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.86.
  34. Michael Pinsker. Current challenges in infinite-domain constraint satisfaction: Dilemmas of the infinite sheep. In 2022 IEEE 52nd International Symposium on Multiple-Valued Logic (ISMVL), pages 80-87, Los Alamitos, CA, USA, 2022. IEEE Computer Society. URL: https://doi.org/10.1109/ISMVL52857.2022.00019.
  35. Emil L. Post. The two-valued iterative systems of mathematical logic. Annals of Mathematics Studies, 5, 1941. Google Scholar
  36. Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the tenth annual ACM Symposium on Theory of Computing (STOC '78), pages 216-226, New York, NY, USA, 1978. Association for Computing Machinery. URL: https://doi.org/10.1145/800133.804350.
  37. Simon Thomas. Reducts of the random graph. The Journal of Symbolic Logic, 56(1):176-181, 1991. URL: https://doi.org/10.2307/2274912.
  38. Simon Thomas. Reducts of random hypergraphs. Annals of Pure and Applied Logic, 80(2):165-193, 1996. URL: https://doi.org/10.1016/0168-0072(95)00061-5.
  39. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
  40. Dmitriy Zhuk. A proof of the CSP dichotomy conjecture. Journal of the ACM, 67(5):30:1-30:78, 2020. 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