A Definability Dichotomy for Finite Valued CSPs

Authors Anuj Dawar, Pengming Wang

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Anuj Dawar
Pengming Wang

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Anuj Dawar and Pengming Wang. A Definability Dichotomy for Finite Valued CSPs. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 60-77, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


Finite valued constraint satisfaction problems are a formalism for describing many natural optimisation problems, where constraints on the values that variables can take come with rational weights and the aim is to find an assignment of minimal cost. Thapper and Zivny have recently established a complexity dichotomy for valued constraint languages. They show that each such languages either gives rise to a polynomial-time solvable optimisation problem, or to an NP-hard one, and establish a criterion to distinguish the two cases. We refine the dichotomy by showing that all optimisation problems in the first class are definable in fixed-point language with counting, while all languages in the second class are not definable, even in infinitary logic with counting. Our definability dichotomy is not conditional on any complexity-theoretic assumption.
  • descriptive complexity
  • constraint satisfaction
  • definability
  • fixed-point logic
  • optimization


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  1. M. Anderson, A. Dawar, and B. Holm. Maximum matching and linear programming in fixed-point logic with counting. In Proceedings of the 28th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 173-182, 2013. Google Scholar
  2. A. Atserias, A. Bulatov, and A. Dawar. Affine systems of equations and counting infinitary logic. Theoretical Computer Science, 410(18):1666-1683, 2009. Google Scholar
  3. L. Barto and M. Kozik. Constraint satisfaction problems solvable by local consistency methods. J. ACM, 61, 2014. Google Scholar
  4. A. Bulatov, P. Jeavons, and A. Krokhin. Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34(3):720-742, 2005. Google Scholar
  5. D. Cohen, M.C. Cooper, P. Jeavons, and A. Krokhin. The complexity of soft constraint satisfaction. Artificial Intelligence, 170(11):983-1016, 2006. Google Scholar
  6. A. Dawar. The nature and power of fixed-point logic with counting. ACM SIGLOG News, 2:8-21, 2015. Google Scholar
  7. H-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer, 2nd edition, 1999. Google Scholar
  8. T. Feder and M.Y. Vardi. Computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing, 28:57-104, 1998. Google Scholar
  9. A. Huber, A. Krokhin, and R. Powell. Skew bisubmodularity and valued CSPs. SIAM Journal on Computing, 43(3):1064-1084, 2014. Google Scholar
  10. L. Libkin. Elements of Finite Model Theory. Springer, 2004. Google Scholar
  11. M. Otto. Bounded Variable Logics and Counting - A Study in Finite Models, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997. Google Scholar
  12. J. Thapper and S. Živný. The complexity of finite-valued CSPs. In Proceedings of the 45th ACM Symposium on the Theory of Computing, STOC'13, pages 695-704. ACM, 2013. Google Scholar
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