Document Open Access Logo

The Network Satisfaction Problem for Relation Algebras with at Most 4 Atoms

Authors Manuel Bodirsky , Moritz Jahn , Simon Knäuer , Matěj Konečný , Paul Winkler

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2026.168.pdf
  • Filesize: 0.91 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Constraint and logic programming
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Constraint Satisfaction
  • Computational Complexity
  • Relation Algebras
  • Network Satisfaction
  • Normal Representations
  • Polynomial-Time Algorithms

Metrics

Abstract

Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable [Hajnal Andréka and Roger D. Maddux, 1994]. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard [Matteo Cristiani and Robin Hirsch, 2004]. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.

Cite As Get BibTex

Manuel Bodirsky, Moritz Jahn, Simon Knäuer, Matěj Konečný, and Paul Winkler. The Network Satisfaction Problem for Relation Algebras with at Most 4 Atoms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 168:1-168:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.168

Author Details

Manuel Bodirsky
  • Institut für Algebra, TU Dresden, Germany
Moritz Jahn
  • Institut für Algebra, TU Dresden, Germany
Simon Knäuer
  • Institut für Algebra, TU Dresden, Germany
Matěj Konečný
  • Institut für Algebra, TU Dresden, Germany
Paul Winkler
  • Institut für Algebra, TU Dresden, Germany

Funding

Manuel Bodirsky, Moritz Jahn, Matěj Konečný, and Paul Winkler received funding from the ERC (Grant Agreement no. 101071674, POCOCOP). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency.

Supplementary Materials

References

  1. Hajnal Andréka and Roger D. Maddux. Representations for small relation algebras. Notre Dame Journal of Formal Logic, 35(4):550-562, 1994. URL: https://doi.org/10.1305/NDJFL/1040408612.
  2. Jørgen Bang-Jensen and Jing Huang. Quasi-transitive digraphs. Journal of Graph Theory, 20(2):141-161, 1995. URL: https://doi.org/10.1002/jgt.3190200205.
  3. 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. Preprint arXiv:1612.07551. URL: https://arxiv.org/abs/1612.07551.
  4. Libor Barto, Jakub Opršal, and Michael Pinsker. The wonderland of reflections. Israel Journal of Mathematics, 223(1):363-398, 2018. Google Scholar
  5. Libor Barto and Michael Pinsker. The algebraic dichotomy conjecture for infinite domain constraint satisfaction problems. In Proceedings of the 31st Annual IEEE Symposium on Logic in Computer Science - LICS'16, pages 615-622, 2016. Preprint arXiv:1602.04353. URL: https://arxiv.org/abs/1602.04353.
  6. Manuel Bodirsky. Finite relation algebras with normal representations. In Relational and Algebraic Methods in Computer Science - 17th International Conference, RAMiCS 2018, Groningen, The Netherlands, October 29 - November 1, 2018, Proceedings, pages 3-17, 2018. URL: https://doi.org/10.1007/978-3-030-02149-8_1.
  7. Manuel Bodirsky. Complexity of Infinite-Domain Constraint Satisfaction. Lecture Notes in Logic (52). Cambridge University Press, Cambridge, United Kingdom; New York, NY, 2021. URL: https://doi.org/10.1017/9781107337534.
  8. Manuel Bodirsky, Santiago Guzmán-Pro, Moritz Jahn, Matěj Konečný, and Paul Winkler. Circular chromatic numbers, balanceability, relation algebras, and network satisfaction problems, 2025. Preprint available under https://arxiv.org/abs/2512.06878. URL: https://doi.org/10.48550/arXiv.2512.06878.
  9. Manuel Bodirsky and Jan Kára. The complexity of temporal constraint satisfaction problems. Journal of the ACM, 57(2):1-41, 2009. An extended abstract appeared in the Proceedings of the Symposium on Theory of Computing (STOC). URL: https://doi.org/10.1145/1667053.1667058.
  10. Manuel Bodirsky and Simon Knäuer. Hardness of network satisfaction for relation algebras with normal representations. In Relational and Algebraic Methods in Computer Science, pages 31-46. Springer International Publishing, 2020. URL: https://doi.org/10.1007/978-3-030-43520-2_3.
  11. Manuel Bodirsky and Simon Knäuer. The complexity of network satisfaction problems for symmetric relation algebras with a flexible atom. Journal of Artificial Intelligence Research, 75, 2022. URL: https://doi.org/10.1613/jair.1.14195.
  12. Manuel Bodirsky and Simon Knäuer. Network satisfaction problems solved by k-consistency. In 50th International Colloquium on Automata, Languages, and Programming, ICALP 2023, July 10-14, 2023, Paderborn, Germany, pages 116:1-116:20, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.116.
  13. Manuel Bodirsky and Martin Kutz. Pure dominance constraints. In Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS), pages 287-298, 2002. URL: https://doi.org/10.1007/3-540-45841-7_23.
  14. 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.
  15. Manuel Bodirsky and Michael Pinsker. Schaefer’s theorem for graphs. Journal of the ACM, 62(3):52 pages (article number 19), 2015. A conference version appeared in the Proceedings of STOC 2011, pages 655-664. URL: https://doi.org/10.1145/2764899.
  16. 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.
  17. Mathias Broxvall and Peter Jonsson. Point algebras for temporal reasoning: Algorithms and complexity. Artificial Intelligence, 149(2):179-220, 2003. URL: https://doi.org/10.1016/S0004-3702(03)00075-4.
  18. 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, pages 319-330, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  19. Jakub Bulín, Andrei A. Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 602-613, 2019. URL: https://doi.org/10.1145/3313276.3316300.
  20. Stanley N. Burris and Hanamantagouda P. Sankappanavar. A Course in Universal Algebra. Springer Verlag, Berlin, 1981. Google Scholar
  21. Peter J. Cameron. The random graph. R. L. Graham and J. Nešetřil, Editors, The Mathematics of Paul Erdös, 1996. Google Scholar
  22. Gregory Cherlin. Homogeneous ordered graphs and metrically homogeneous graphs, 2020. Preprint. Google Scholar
  23. Gregory L. Cherlin. The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments. AMS Memoir, 131(621), January 1998. Google Scholar
  24. Stephen D. Comer. Extension of polygroups by polygroups and their representations using color schemes. In Ralph S. Freese and Octavio C. Garcia, editors, Universal Algebra and Lattice Theory, pages 91-103, Berlin, Heidelberg, 1983. Springer Berlin Heidelberg. Google Scholar
  25. Stephen D. Comer. A remark on chromatic polygroups. Congressus Numerantium, 38:85-95, 1983. Google Scholar
  26. Matteo Cristiani and Robin Hirsch. The complexity of the constraint satisfaction problem for small relation algebras. Artificial Intelligence Journal, 156:177-196, 2004. URL: https://doi.org/10.1016/J.ARTINT.2004.02.003.
  27. Michael Garey and David Johnson. A guide to NP-completeness. CSLI Press, Stanford, 1978. Google Scholar
  28. C. Ward Henson. Countable homogeneous relational systems and categorical theories. Journal of Symbolic Logic, 37:494-500, 1972. Google Scholar
  29. R. Hirsch and I. Hodkinson. Representability is not decidable for finite relation algebras. Transactions of the American Mathematical Society, 353(4):1387-1401, 2001. Google Scholar
  30. Robin Hirsch. Relation algebras of intervals. Artificial Intelligence Journal, 83:1-29, 1996. Google Scholar
  31. Robin Hirsch. A finite relation algebra with undecidable network satisfaction problem. Logic Journal of the IGPL, 7(4):547-554, 1999. URL: https://doi.org/10.1093/JIGPAL/7.4.547.
  32. Wilfrid Hodges. A shorter model theory. Cambridge University Press, Cambridge, 1997. Google Scholar
  33. Alistair H. Lachlan and Robert E. Woodrow. Countable ultrahomogeneous undirected graphs. Transactions of the AMS, 262(1):51-94, 1980. Google Scholar
  34. Richard E. Ladner. On the structure of polynomial time reducibility. Journal of the ACM, 22(1):155-171, 1975. URL: https://doi.org/10.1145/321864.321877.
  35. R. Lyndon. The representation of relational algebras. Annals of Mathematics, 51(3):707-729, 1950. Google Scholar
  36. Roger D. Maddux. Finite symmetric integral relation algebras with no 3-cycles. In Renate A. Schmidt, editor, Relations and Kleene Algebra in Computer Science, 9th International Conference on Relational Methods in Computer Science and 4th International Workshop on Applications of Kleene Algebra, RelMiCS/AKA 2006, Manchester, UK, August 29-September 2, 2006, Proceedings, volume 4136 of Lecture Notes in Computer Science, pages 2-29. Springer, 2006. URL: https://doi.org/10.1007/11828563_2.
  37. Roger Duncan Maddux. Relation Algebras: Volume 150. Studies in logic and the foundations of mathematics. Elsevier Science, London, England, May 2006. Google Scholar
  38. Donald Monk. On representable relation algebras. Michigan Mathematical Journal, 11(3):207-210, 1964. URL: https://doi.org/10.1307/mmj/1028999131.
  39. Alfred Tarski. On the calculus of relations. J. Symbolic Logic, 6, 1941. URL: https://doi.org/10.2307/2268577.
  40. Alfred Tarski. Contributions to the theory of models. Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, 58:56-64, 1955. Google Scholar
  41. Marc Vilain, Henry Kautz, and Peter van Beek. Constraint propagation algorithms for temporal reasoning: A revised report. Reading in Qualitative Reasoning about Physical Systems, pages 373-381, 1989. Google Scholar
  42. 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.
  43. Dmitriy N. 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, pages 331-342, 2017. URL: https://arxiv.org/abs/1704.01914.
  44. Paul-Hermann Zieschang. Hypergroups. Springer, 2023. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact