Document Open Access Logo

Promise and Infinite-Domain Constraint Satisfaction

Author Antoine Mottet



PDF
Thumbnail PDF

File

LIPIcs.CSL.2024.41.pdf
  • Filesize: 0.75 MB
  • 19 pages

Document Identifiers

Author Details

Antoine Mottet
  • Hamburg University of Technology, Research Group for Theoretical Computer Science, Germany

Cite AsGet BibTex

Antoine Mottet. Promise and Infinite-Domain Constraint Satisfaction. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CSL.2024.41

Abstract

Two particularly active branches of research in constraint satisfaction are the study of promise constraint satisfaction problems (PCSPs) with finite templates and the study of infinite-domain constraint satisfaction problems with ω-categorical templates. In this paper, we explore some connections between these two hitherto unrelated fields and describe a general approach to studying the complexity of PCSPs by constructing suitable infinite CSP templates. As a result, we obtain new characterizations of the power of various classes of algorithms for PCSPs, such as first-order logic, arc consistency reductions, and obtain new proofs of the characterizations of the power of the classical linear and affine relaxations for PCSPs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
Keywords
  • promise constraint satisfaction problems
  • polymorphisms
  • homogeneous structures
  • first-order logic

Metrics

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

References

  1. Kristina Asimi and Libor Barto. Finitely tractable promise constraint satisfaction problems. In Filippo Bonchi and Simon J. Puglisi, editors, 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, August 23-27, 2021, Tallinn, Estonia, volume 202 of LIPIcs, pages 11:1-11:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.11.
  2. Albert Atserias. On digraph coloring problems and treewidth duality. Eur. J. Comb., 29(4):796-820, 2008. URL: https://doi.org/10.1016/j.ejc.2007.11.004.
  3. Albert Atserias, Andrei A. Bulatov, and Anuj Dawar. Affine systems of equations and counting infinitary logic. Theor. Comput. Sci., 410(18):1666-1683, 2009. URL: https://doi.org/10.1016/j.tcs.2008.12.049.
  4. Albert Atserias and Víctor Dalmau. Promise constraint satisfaction and width. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9-12, 2022, pages 1129-1153. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.48.
  5. Albert Atserias and Szymon Toruńczyk. Non-homogenizable classes of finite structures. In Jean-Marc Talbot and Laurent Regnier, editors, 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29 - September 1, 2016, Marseille, France, volume 62 of LIPIcs, pages 16:1-16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CSL.2016.16.
  6. Per Austrin, Venkatesan Guruswami, and Johan Håstad. (2+ε)-Sat is NP-hard. SIAM J. Comput., 46(5):1554-1573, 2017. URL: https://doi.org/10.1137/15M1006507.
  7. Libor Barto. Promises make finite (constraint satisfaction) problems infinitary. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1-8. IEEE, 2019. URL: https://doi.org/10.1109/LICS.2019.8785671.
  8. Libor Barto, Jakub Bulin, Andrei A. Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. J. ACM, 68(4):28:1-28:66, 2021. URL: https://doi.org/10.1145/3457606.
  9. Libor Barto and Marcin Kozik. Constraint satisfaction problems solvable by local consistency methods. J. ACM, 61(1):3:1-3:19, 2014. URL: https://doi.org/10.1145/2556646.
  10. Libor Barto and Marcin Kozik. Combinatorial gap theorem and reductions between promise CSPs. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9-12, 2022, pages 1204-1220. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.50.
  11. Manuel Bodirsky and Martin Grohe. Non-dichotomies in constraint satisfaction complexity. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part II - Track B: Logic, Semantics, and Theory of Programming & Track C: Security and Cryptography Foundations, volume 5126 of Lecture Notes in Computer Science, pages 184-196. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-70583-3_16.
  12. Manuel Bodirsky, Martin Hils, and Barnaby Martin. On the scope of the universal-algebraic approach to constraint satisfaction. In Proceedings of the 25th Annual IEEE Symposium on Logic in Computer Science, LICS 2010, 11-14 July 2010, Edinburgh, United Kingdom, pages 90-99. IEEE Computer Society, 2010. URL: https://doi.org/10.1109/LICS.2010.13.
  13. Manuel Bodirsky, Florent R. Madelaine, and Antoine Mottet. A proof of the algebraic tractability conjecture for monotone monadic SNP. SIAM J. Comput., 50(4):1359-1409, 2021. URL: https://doi.org/10.1137/19M128466X.
  14. Manuel Bodirsky, Antoine Mottet, Miroslav Olsak, Jakub Opršal, Michael Pinsker, and Ross Willard. ω-categorical structures avoiding height 1 identities. Trans. Amer. Math. Soc., 374:327-350, 2021. Google Scholar
  15. Manuel Bodirsky, Antoine Mottet, Miroslav Olšák, Jakub Opršal, Michael Pinsker, and Ross Willard. Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems). In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1-12. IEEE, 2019. URL: https://doi.org/10.1109/LICS.2019.8785883.
  16. Manuel Bodirsky, Michael Pinsker, and Todor Tsankov. Decidability of definability. The Journal of Symbolic Logic, 78(4):1036-1054, 2013. URL: https://doi.org/10.2178/jsl.7804020.
  17. Joshua Brakensiek and Venkatesan Guruswami. An algorithmic blend of LPs and ring equations for promise CSPs. 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 436-455. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.28.
  18. Joshua Brakensiek and Venkatesan Guruswami. Promise constraint satisfaction: Algebraic structure and a symmetric boolean dichotomy. SIAM J. Comput., 50(6):1663-1700, 2021. URL: https://doi.org/10.1137/19M128212X.
  19. Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, and Stanislav Zivný. The power of the combined basic linear programming and affine relaxation for promise constraint satisfaction problems. SIAM J. Comput., 49(6):1232-1248, 2020. URL: https://doi.org/10.1137/20M1312745.
  20. Raimundo Briceno, Andrei Bulatov, Víctor Dalmau, and Benoit Larose. Dismantlability, connectedness, and mixing in relational structures. J. Comb. Theory, Ser. B, 147:37-70, 2021. URL: https://doi.org/10.1016/j.jctb.2020.10.001.
  21. 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.
  22. Gregory Cherlin, Saharon Shelah, and Niandong Shi. Universal graphs with forbidden subgraphs and algebraic closure. Advances in Applied Mathematics, 22:454-491, 1999. Google Scholar
  23. Lorenzo Ciardo and Stanislav Živný. CLAP: A new algorithm for promise CSPs. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9-12, 2022, pages 1057-1068. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.46.
  24. Lorenzo Ciardo and Stanislav Zivný. Hierarchies of minion tests for PCSPs through tensors. CoRR, abs/2207.02277, 2022. URL: https://doi.org/10.48550/arXiv.2207.02277.
  25. Lorenzo Ciardo and Stanislav Zivný. Approximate graph colouring and crystals. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 2256-2267. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH86.
  26. Lorenzo Ciardo and Stanislav Zivný. Approximate graph colouring and the hollow shadow. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 623-631. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585112.
  27. Adam Ó Conghaile. Cohomology in constraint satisfaction and structure isomorphism. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022, August 22-26, 2022, Vienna, Austria, volume 241 of LIPIcs, pages 75:1-75:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.75.
  28. Víctor Dalmau and Jakub Opršal. Local consistency as a reduction between constraint satisfaction problems. CoRR, abs/2301.05084, 2023. URL: https://doi.org/10.48550/arXiv.2301.05084.
  29. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput., 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  30. Pierre Gillibert, Julius Jonusas, Michael Kompatscher, Antoine Mottet, and Michael Pinsker. Hrushovski’s encoding and ω-categorical CSP monsters. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 131:1-131:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.131.
  31. Pierre Gillibert, Julius Jonusas, Michael Kompatscher, Antoine Mottet, and Michael Pinsker. When symmetries are not enough: A hierarchy of hard constraint satisfaction problems. SIAM J. Comput., 51(2):175-213, 2022. URL: https://doi.org/10.1137/20m1383471.
  32. Jan Hubička and Jaroslav Nešetřil. All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms). CoRR, abs/1606.07979, 2016. URL: https://arxiv.org/abs/1606.07979.
  33. Andrei A. Krokhin and Jakub Opršal. An invitation to the promise constraint satisfaction problem. ACM SIGLOG News, 9(3):30-59, 2022. URL: https://doi.org/10.1145/3559736.3559740.
  34. Andrei A. Krokhin, Jakub Opršal, Marcin Wrochna, and Stanislav Zivný. Topology and adjunction in promise constraint satisfaction. SIAM J. Comput., 52(1):38-79, 2023. URL: https://doi.org/10.1137/20M1378223.
  35. Gábor Kun, Ryan O'Donnell, Suguru Tamaki, Yuichi Yoshida, and Yuan Zhou. Linear programming, width-1 CSPs, and robust satisfaction. In Shafi Goldwasser, editor, Innovations in Theoretical Computer Science 2012, Cambridge, MA, USA, January 8-10, 2012, pages 484-495. ACM, 2012. URL: https://doi.org/10.1145/2090236.2090274.
  36. Benoit Larose, Cynthia Loten, and Claude Tardif. A characterisation of first-order constraint satisfaction problems. Log. Methods Comput. Sci., 3(4), 2007. URL: https://doi.org/10.2168/LMCS-3(4:6)2007.
  37. Antoine Mottet and Michael Pinsker. Smooth approximations and CSPs over finitely bounded homogeneous structures. In Christel Baier and Dana Fisman, editors, LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science, Haifa, Israel, August 2-5, 2022, pages 36:1-36:13. ACM, 2022. URL: https://doi.org/10.1145/3531130.3533353.
  38. Jaroslav Nešetřil and Claude Tardif. Duality theorems for finite structures (characterising gaps and good characterisations). Journal of Combinatorial Theory, Series B, 80(1):80-97, 2000. URL: https://doi.org/10.1006/jctb.2000.1970.
  39. Michael Pinsker. Current challenges in infinite-domain constraint satisfaction: Dilemmas of the infinite sheep. In 52nd IEEE International Symposium on Multiple-Valued Logic, ISMVL 2022, Dallas, TX, USA, May 18-20, 2022, pages 80-87. IEEE, 2022. URL: https://doi.org/10.1109/ISMVL52857.2022.00019.
  40. Michael Pinsker and Manuel Bodirsky. Canonical functions: a proof via topological dynamics. Contributions Discret. Math., 16(2):36-45, 2021. URL: https://cdm.ucalgary.ca/article/view/71724.
  41. Benjamin Rossman. Homomorphism preservation theorems. J. ACM, 55(3):15:1-15:53, 2008. URL: https://doi.org/10.1145/1379759.1379763.
  42. Marcin Wrochna and Stanislav Živný. Improved hardness for H-colourings of G-colourable 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 1426-1435. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.86.
  43. 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.
  44. 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.
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