Sensitive Instances of the Constraint Satisfaction Problem

Authors Libor Barto , Marcin Kozik , Johnson Tan , Matt Valeriote

Thumbnail PDF


  • Filesize: 0.55 MB
  • 18 pages

Document Identifiers

Author Details

Libor Barto
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Praha 8, Czech Republic
Marcin Kozik
  • Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Johnson Tan
  • Department of Mathematics, University of Illinois, Urbana-Champaign, Urbana, IL, USA
Matt Valeriote
  • Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada

Cite AsGet BibTex

Libor Barto, Marcin Kozik, Johnson Tan, and Matt Valeriote. Sensitive Instances of the Constraint Satisfaction Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 110:1-110:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We investigate the impact of modifying the constraining relations of a Constraint Satisfaction Problem (CSP) instance, with a fixed template, on the set of solutions of the instance. More precisely we investigate sensitive instances: an instance of the CSP is called sensitive, if removing any tuple from any constraining relation invalidates some solution of the instance. Equivalently, one could require that every tuple from any one of its constraints extends to a solution of the instance. Clearly, any non-trivial template has instances which are not sensitive. Therefore we follow the direction proposed (in the context of strict width) by Feder and Vardi in [Feder and Vardi, 1999] and require that only the instances produced by a local consistency checking algorithm are sensitive. In the language of the algebraic approach to the CSP we show that a finite idempotent algebra 𝔸 has a k+2 variable near unanimity term operation if and only if any instance that results from running the (k, k+1)-consistency algorithm on an instance over 𝔸² is sensitive. A version of our result, without idempotency but with the sensitivity condition holding in a variety of algebras, settles a question posed by G. Bergman about systems of projections of algebras that arise from some subalgebra of a finite product of algebras. Our results hold for infinite (albeit in the case of 𝔸 idempotent) algebras as well and exhibit a surprising similarity to the strict width k condition proposed by Feder and Vardi. Both conditions can be characterized by the existence of a near unanimity operation, but the arities of the operations differ by 1.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Constraint and logic programming
  • Constraint satisfaction problem
  • bounded width
  • local consistency
  • near unanimity operation
  • loop lemma


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


  1. Kirby A. Baker and Alden F. Pixley. Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z., 143(2):165-174, 1975. URL:
  2. Libor Barto. Finitely related algebras in congruence distributive varieties have near unanimity terms. Canad. J. Math., 65(1):3-21, 2013. URL:
  3. Libor Barto. The collapse of the bounded width hierarchy. J. Logic Comput., 26(3):923-943, 2016. URL:
  4. Libor Barto and Marcin Kozik. Congruence distributivity implies bounded width. SIAM J. Comput., 39(4):1531–1542, December 2009. URL:
  5. Libor Barto and Marcin Kozik. Constraint satisfaction problems solvable by local consistency methods. J. ACM, 61(1):3:1-3:19, January 2014. URL:
  6. Libor Barto and Marcin Kozik. Absorption in Universal Algebra and CSP. In Andrei Krokhin and Stanislav Zivny, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 45-77. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2017. URL:
  7. Libor Barto, Andrei Krokhin, and Ross Willard. Polymorphisms, and How to Use Them. In Andrei Krokhin and Stanislav Zivny, editors, The Constraint Satisfaction Problem: Complexity and Approximability, volume 7 of Dagstuhl Follow-Ups, pages 1-44. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2017. URL:
  8. George M. Bergman. On the existence of subalgebras of direct products with prescribed d-fold projections. Algebra Universalis, 7(3):341-356, 1977. URL:
  9. Manuel Bodirsky. Complexity classification in infinite-domain constraint satisfaction. Mémoire d'habilitation à diriger des recherches, Université Diderot - Paris 7, 2012. URL:
  10. Manuel Bodirsky and Víctor Dalmau. Datalog and constraint satisfaction with infinite templates. Journal of Computer and System Sciences, 79(1):79-100, 2013. URL:
  11. Andrei A. Bulatov. Bounded relational width. Preprint, 2009. Google Scholar
  12. H. Chen, M. Valeriote, and Y. Yoshida. Testing assignments to constraint satisfaction problems. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 525-534, October 2016. URL:
  13. 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 (electronic), 1999. URL:
  14. András Huhn. Schwach distributive Verbände. Acta Fac. Rerum Natur. Univ. Comenian. Math., pages 51-56, 1971. Google Scholar
  15. Peter Jeavons, David Cohen, and Martin C. Cooper. Constraints, consistency and closure. Artificial Intelligence, 101(1-2):251-265, 1998. URL:
  16. Emil W. Kiss and Péter Pröhle. Problems and results in tame congruence theory. A survey of the '88 Budapest Workshop. Algebra Universalis, 29(2):151-171, 1992. URL:
  17. Marcin Kozik, Andrei Krokhin, Matt Valeriote, and Ross Willard. Characterizations of several Maltsev conditions. Algebra Universalis, 73(3-4):205-224, 2015. URL:
  18. Miroslav Olšák. The weakest nontrivial idempotent equations. Bulletin of the London Mathematical Society, 49(6):1028-1047, 2017. URL:
  19. Michael Pinsker. Algebraic and model theoretic methods in constraint satisfaction, 2015. URL:
  20. Ágnes Szendrei. Rosenberg-type completeness criteria for subclones of Slupecki’s clone. Proceedings of The International Symposium on Multiple-Valued Logic, pages 349-354, 2012. URL: