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Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width

Author Michał Wrona

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LIPIcs.STACS.2020.39.pdf
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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Constraint Satisfaction
  • Homogeneous Graphs
  • Bounded Width
  • Strict Width
  • Relational Width
  • Computational Complexity

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Abstract

Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ℋ := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking.
Our main result is that the structures ? under consideration have relational width exactly (2, ?_ℋ) where ?_ℋ is the maximal size of a forbidden subgraph of ℋ, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_ℋ may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3).

Cite As Get BibTex

Michał Wrona. Relational Width of First-Order Expansions of Homogeneous Graphs with Bounded Strict Width. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.39

Author Details

Michał Wrona
  • Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Poland

Funding

The research was partially supported by NCN grant number 2014/14/A/ST6/00138.

References

  1. Libor Barto. The collapse of the bounded width hierarchy. J. Log. Comput., 26(3):923-943, 2016. Google Scholar
  2. Libor Barto and Marcin Kozik. Constraint satisfaction problems solvable by local consistency methods. J. ACM, 61(1):3:1-3:19, 2014. An extended abstract appeared in the Proceedings of the Symposium on Foundations of Computer Science (FOCS'09). URL: https://doi.org/10.1145/2556646.
  3. Libor Barto and Michael Pinsker. The algebraic dichotomy conjecture for infinite domain constraint satisfaction problems. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, (LICS), pages 615-622, 2016. Google Scholar
  4. Manuel Bodirsky and Hubert Chen. Oligomorphic clones. Algebra Universalis, 57(1):109-125, 2007. Google Scholar
  5. Manuel Bodirsky and Víctor Dalmau. Datalog and constraint satisfaction with infinite templates. J. Comput. Syst. Sci., 79(1):79-100, 2013. URL: https://doi.org/10.1016/j.jcss.2012.05.012.
  6. Manuel Bodirsky and Peter Jonsson. A model-theoretic view on qualitative constraint reasoning. J. Artif. Intell. Res. (JAIR), 58:339-385, 2017. Google Scholar
  7. Manuel Bodirsky and Jan Kára. The complexity of equality constraint languages. Theory Comput. Syst., 43(2):136-158, 2008. Google Scholar
  8. 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'08). Google Scholar
  9. Manuel Bodirsky, Barnaby Martin, Michael Pinsker, and András Pongrácz. Constraint satisfaction problems for reducts of homogeneous graphs. CoRR, abs/1602.05819, 2016. An extended abstract appeared in the Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP'16). Google Scholar
  10. Manuel Bodirsky and Antoine Mottet. A dichotomy for first-order reducts of unary structures. Logical Methods in Computer Science, 14(2), 2018. A preliminary version of the paper appeared in the proceedings of LICS'16. Google Scholar
  11. Manuel Bodirsky and Jaroslav Nesetril. Constraint satisfaction with countable homogeneous templates. J. Log. Comput., 16(3):359-373, 2006. Google Scholar
  12. Manuel Bodirsky and Michael Pinsker. Reducts of Ramsey structures. AMS Contemporary Mathematics, vol. 558 (Model Theoretic Methods in Finite Combinatorics), pages 489-519, 2011. Google Scholar
  13. Manuel Bodirsky and Michael Pinsker. Schaefer’s theorem for graphs. J. ACM, 62(3):19:1-19:52, 2015. An extended abstract appeared in the Proceedings of the Symposium on Theory of Computing (STOC'11). Google Scholar
  14. Manuel Bodirsky and MichałWrona. Equivalence constraint satisfaction problems. In Proceedings of Computer Science Logic, volume 16 of LIPICS, pages 122-136. Dagstuhl Publishing, September 2012. Google Scholar
  15. Andrei A. Bulatov. Bounded relational width. manuscript, 2009. Google Scholar
  16. Andrei A. Bulatov. A dichotomy theorem for nonuniform csps. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 319-330. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  17. Andrei A. Bulatov, Andrei A. Krokhin, and Peter G. Jeavons. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34:720-742, 2005. A preliminary version of the paper appeared in the proceedings of the Symposium on Logic in Computer Science (ICALP'00). Google Scholar
  18. Peter J. Cameron. Oligomorphic Permutation Groups. Cambridge University Press, Cambridge, 1990. Google Scholar
  19. Rina Dechter. From local to global consistency. Artificial Intelligence, 55(1):87-108, 1992. Google Scholar
  20. 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, 28:57-104, 1999. A preliminary version appeared in the proceedings of the Symposium on Theory of Computing (STOC'93). Google Scholar
  21. Wilfrid Hodges. A shorter model theory. Cambridge University Press, Cambridge, 1997. Google Scholar
  22. Peter Jeavons, David Cohen, and Marc Gyssens. Closure properties of constraints. Journal of the ACM, 44(4):527-548, 1997. Google Scholar
  23. Michael Kompatscher and Van Trung Pham. A complexity dichotomy for poset constraint satisfaction. In 34th Symposium on Theoretical Aspects of Computer Science, STACS'17, pages 47:1-47:12, 2017. Google Scholar
  24. A. H. Lachlan and Robert E. Woodrow. Countable ultrahomogeneous undirected graphs. Trans. Amer. Math. Soc., 262:51-94, 1980. Google Scholar
  25. Jochen Renz. Implicit constraints for qualitative spatial and temporal reasoning. In Principles of Knowledge Representation and Reasoning: Proceedings of the Thirteenth International Conference, KR'12, 2012. Google Scholar
  26. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 331-342. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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