article

A dichotomy theorem for constraint satisfaction problems on a 3-element set

Author:

Andrei A. BulatovAuthors Info & Claims

Journal of the ACM (JACM), Volume 53, Issue 1

Pages 66 - 120

https://doi.org/10.1145/1120582.1120584

Published: 01 January 2006 Publication History

Abstract

The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate those subclasses of the CSP that are tractable and those which remain NP-complete.Schaefer gave an exhaustive solution of this problem for the CSP on a 2-element domain. In this article, we generalise this result to a classification of the complexity of the CSP on a 3-element domain. The main result states that every subproblem of the CSP is either tractable or NP-complete, and the criterion separating them is that conjectured in Bulatov et al. [2005] and Bulatov and Jeavons [2001b]. We also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure. Finally, we exhibit a polynomial time algorithm which, for a given set of allowed constraints, outputs if this set gives rise to a tractable problem class. To obtain the main result and the algorithm, we extensively use the algebraic technique for the CSP developed in Jeavons [1998b], Bulatov et al.[2005], and Bulatov and Jeavons [2001b].

References

[1]

Allen, J. 1994. Natural Language Understanding. Benjamin Cummings.]]

[2]

Bulatov, A. 2002. Mal'tsev constraints are tractable. Tech. Rep. PRG-RR-02-05, Computing Laboratory, University of Oxford, Oxford, UK.]]

[3]

Bulatov, A. 2006a. Combinatorial problems raised from 2-semilattices. J. Alg., submitted for publication.]]

[4]

Bulatov, A. 2006b. Three-element Mal'tsev algebras. Acta Sci. Math (Szeged). To appear.]]

[5]

Bulatov, A., and Jeavons, P. 2000. Tractable constraints closed under a binary operation. Tech. Rep. PRG-TR-12-00, Computing Laboratory, University of Oxford, Oxford, UK.]]

[6]

Bulatov, A., and Jeavons, P. 2001a. Algebraic approach to multisorted constraints. Tech. Rep. PRG-RR-01-18, Computing Laboratory, University of Oxford, Oxford, UK.]]

[7]

Bulatov, A., and Jeavons, P. 2001b. Algebraic structures in combinatorial problems. Tech. Rep. MATH-AL-4-2001, Technische universität Dresden, Dresden, Germany.]]

[8]

Bulatov, A., and Jeavons, P. 2003. An algebraic approach to multisorted constraits. In Proceedings of the 9th International Conference on Principles and Practice of Constraint Programming (CP'03) (Kinsale, Ireland). 197--202.]]

[9]

Bulatov, A., Jeavons, P., and Krokhin, A. 2001. The complexity of maximal constraint languages. In Proceedings of the 33rd Annual ACM Simposium on Theory of Computing (Hersonissos, Crete, Greece). ACM, New York, 667--674.]]

Digital Library

[10]

Bulatov, A., Jeavons, P., and Krokhin, A. 2005. Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720--742.]]

Digital Library

[11]

Burris, S., and Sankappanavar, H. 1981. A Course in Universal Algebra. Graduate Texts in Mathematics, vol. 78. Springer-Verlag, New York-Berlin.]]

[12]

Cooper, M. 1989. An optimal k-consistency algorithm. Artif. Intel. 41, 89--95.]]

Digital Library

[13]

Creignou, N., Khanna, S., and Sudan, M. 2001. Complexity Classifications of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia, PA.]]

Digital Library

[14]

Dalmau, V. 2000. Computational complexity of problems over generalized formulas. Ph.D. dissertation, Department LSI of the Universitat Politecnica de Catalunya (UPC), Barcelona, Spain.]]

[15]

Dechter, R. 1992. From local to global consistency. Artif. Intel. 55, 1, 87--107.]]

Digital Library

[16]

Dechter, R. 2003. Constraint processing. Morgan-Kaufmann, San Francisco, CA.]]

Digital Library

[17]

Dechter, R., and Dechter, A. 1996. Structure-driven algorithms for truth maintenance. Artif. Intel. 82, 1--2, 1--20.]]

Digital Library

[18]

Dechter, R., and Meiri, I. 1994. Experimental evaluation of preprocessing algorithms for constraint satisfaction problems. Artif. Intel. 68, 211--241.]]

Digital Library

[19]

Dechter, R., and Pearl, J. 1988. Network-based heuristics for constraint satisfaction problems. Artif. Intel. 34, 1, 1--38.]]

Digital Library

[20]

Dechter, R., and van Beek, P. 1997. Local and global relational consistency. Theoret. Comput. Sci. 173, 1, 283--308.]]

Digital Library

[21]

Denecke, K., and Wismath, S. 2002. Universal Algebra and Applications in Theoretical Computer Science. Chapman and Hall/CRC Press.]]

Digital Library

[22]

Feder, T., and Vardi, M. 1998. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput. 28, 57--104.]]

Digital Library

[23]

Freuder, E. 1982. A sufficient condition for backtrack-free search. J. ACM 29, 1, 24--32.]]

Digital Library

[24]

Gottlob, G., Leone, L., and Scarcello, F. 2000. A comparison of structural CSP decomposition methods. Artif. Intel. 124, 2, 243--282.]]

Digital Library

[25]

Jeavons, P. 1998a. Constructing constraints. In Proceedings 4th International Conference on Constraint Programming---CP'98 (Pisa, Oct.). Lecture Notes in Computer Science, vol. 1520. Springer-Verlag, New York, 2--16.]]

Digital Library

[26]

Jeavons, P. 1998b. On the algebraic structure of combinatorial problems. Theoret. Comput. Sci. 200, 185--204.]]

Digital Library

[27]

Jeavons, P., Cohen, D., and Cooper, M. 1998a. Constraints, consistency and closure. Artif. Intel. 101, 1--2, 251--265.]]

[28]

Jeavons, P., Cohen, D., and Gyssens, M. 1997. Closure properties of constraints. J. ACM 44, 527--548.]]

Digital Library

[29]

Jeavons, P., Cohen, D., and Pearson, J. 1998b. Constraints and universal algebra. Ann. Math. Artif. Intel. 24, 51--67.]]

Digital Library

[30]

Kolaitis, P. 2003. Constraint satisfaction, databases, and logic. In Proceedings of the 17th International Joint Conference on Artificial Intellignece (IJCAI'03).]]

Digital Library

[31]

Kolaitis, P., and Vardi, M. 2000a. Conjunctive-query containment and constraint satisfaction. J. Comput. Syst. Sci. 61, 302--332.]]

Digital Library

[32]

Kolaitis, P., and Vardi, M. 2000b. A game-theoretic approach to constraint satisfaction. In Proceedings of the 17th National (US) Conference on Artificial Intelligence (AAAI'00). 175--181.]]

Digital Library

[33]

Kumar, V. 1992. Algorithms for constraint satisfaction problems: A survey. AI Magazine 13, 1, 32--44.]]

Digital Library

[34]

Ladner, R. 1975. On the structure of polynomial time reducibility. J. ACM 22, 155--171.]]

Digital Library

[35]

McKenzie, R., McNulty, G., and Taylor, W. 1987. Algebras, Lattices and Varieties. Vol. I. Wadsworth and Brooks, California.]]

[36]

Montanari, U. 1974. Networks of constraints: Fundamental properties and applications to picture processing. Inf. Sci. 7, 95--132.]]

[37]

Nadel, B. 1995. Constraint satisfaction in Prolog: Complexity and theory-based heuristics. Inf. Sci. 83, 3--4, 113--131.]]

Digital Library

[38]

Nadel, B., and Lin, J. 1991. Automobile transmission design as a constraint satisfaction problem: Modeling the kinematik level. Artif. Intel. Eng. Des. Anal. Manuf. (AI EDAM) 5, 3, 137--171.]]

[39]

Pöschel, R., and Kalužnin, L. 1979. Funktionen- und Relationenalgebren. DVW, Berlin, Germany.]]

[40]

Post, E. 1941. The Two-Valued Iterative Systems of Mathematical Logic. Annals Mathematical Studies, vol. 5. Princeton University Press.]]

[41]

Schaefer, T. 1978. The complexity of satisfiability problems. In Proceedings of the 10th ACM Symposium on Theory of Computing (STOC'78). ACM, New York, 216--226.]]

Digital Library

[42]

Schwalb, E., and Vila, L. 1998. Temporal constraints: A survey. Constraints 3, 2--3, 129--149.]]

Digital Library

[43]

Szendrei, A. 1990. Simple surjective algebras having no proper subalgebras. J. Austral. Math. Soc. (Ser. A) 48, 434--454.]]

[44]

Tsang, E. 1993. Foundations of Constraint Satisfaction. Academic Press, London, England.]]

[45]

van Hentenryck, P., Deville, Y., and Teng, C.-M. 1992. A generic arc-consistency algorithm and its specializations. Artif. Intel. 57, 291--321.]]

Digital Library

[46]

Vardi, M. 2000. Constraint satisfaction and database theory: A tutorial. In Proceedings of 19th ACM Symposium on Priciples of Database Systems (PODS'00). ACM, New York.]]

Digital Library

Cited By

Liu YFan ACai J(2025)Restricted holant dichotomy on domain sizes 3 and 4Theoretical Computer Science10.1016/j.tcs.2024.1149311023(114931)Online publication date: Jan-2025
https://doi.org/10.1016/j.tcs.2024.114931
Yang PHuang YFu Z(2024)A complexity trichotomy for k-regular asymmetric spin systems with complex edge functionsTheoretical Computer Science10.1016/j.tcs.2024.1148351020(114835)Online publication date: Dec-2024
https://doi.org/10.1016/j.tcs.2024.114835
Barsukov AKanté M(2024)Generalisations of matrix partitionsTheoretical Computer Science10.1016/j.tcs.2024.1146521007:COnline publication date: 29-Jul-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114652
Show More Cited By

Index Terms

A dichotomy theorem for constraint satisfaction problems on a 3-element set
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness
  2. Logic
    1. Constraint and logic programming

Recommendations

Complexity of conservative constraint satisfaction problems

In a constraint satisfaction problem (CSP), the aim is to find an assignment of values to a given set of variables, subject to specified constraints. The CSP is known to be NP-complete in general. However, certain restrictions on the form of the allowed ...
A dichotomy theorem for the approximate counting of complex-weighted bounded-degree Boolean CSPs

We determine the computational complexity of approximately counting the total weight of variable assignments for every complex-weighted Boolean constraint satisfaction problem (or CSP) with any number of additional unary (i.e., arity 1) constraints, ...
Approximate counting for complex-weighted Boolean constraint satisfaction problems

Constraint satisfaction problems (or CSPs) have been extensively studied in, for instance, artificial intelligence, database theory, graph theory, and statistical physics. From a practical viewpoint, it is beneficial to approximately solve those CSPs. ...

Comments

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 53, Issue 1

January 2006

206 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/1120582

Issue’s Table of Contents

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 2006

Published in JACM Volume 53, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

287
Total Citations
View Citations
1,154
Total Downloads

Downloads (Last 12 months)21
Downloads (Last 6 weeks)1

Reflects downloads up to 26 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Liu YFan ACai J(2025)Restricted holant dichotomy on domain sizes 3 and 4Theoretical Computer Science10.1016/j.tcs.2024.1149311023(114931)Online publication date: Jan-2025
https://doi.org/10.1016/j.tcs.2024.114931
Yang PHuang YFu Z(2024)A complexity trichotomy for k-regular asymmetric spin systems with complex edge functionsTheoretical Computer Science10.1016/j.tcs.2024.1148351020(114835)Online publication date: Dec-2024
https://doi.org/10.1016/j.tcs.2024.114835
Barsukov AKanté M(2024)Generalisations of matrix partitionsTheoretical Computer Science10.1016/j.tcs.2024.1146521007:COnline publication date: 29-Jul-2024
https://dl.acm.org/doi/10.1016/j.tcs.2024.114652
Bulatov A(2024)Graphs of finite algebras: edges, and connectivityAlgebra universalis10.1007/s00012-024-00865-585:4Online publication date: 30-Oct-2024
https://doi.org/10.1007/s00012-024-00865-5
Ciardo LŽivný S(2023)CLAP: A New Algorithm for Promise CSPsSIAM Journal on Computing10.1137/22M147643552:1(1-37)Online publication date: 25-Jan-2023
https://doi.org/10.1137/22M1476435
Barto LBodor BKozik MMottet APinsker M(2023)Symmetries of Graphs and Structures that Fail to Interpret a Finite Thing2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175732(1-13)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175732
Chatziafratis VMakarychev K(2023)Triplet Reconstruction and all other Phylogenetic CSPs are Approximation Resistant2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00024(253-284)Online publication date: 6-Nov-2023
https://doi.org/10.1109/FOCS57990.2023.00024
Cai JFan ALiu Y(2023)Bipartite 3-regular counting problems with mixed signsJournal of Computer and System Sciences10.1016/j.jcss.2023.01.006135:C(15-31)Online publication date: 1-Aug-2023
https://dl.acm.org/doi/10.1016/j.jcss.2023.01.006
Liu YFan ACai J(2023)Restricted Holant Dichotomy on Domains 3 and 4Combinatorial Optimization and Applications10.1007/978-3-031-49614-1_5(83-96)Online publication date: 15-Dec-2023
https://dl.acm.org/doi/10.1007/978-3-031-49614-1_5
Fujii SIwamasa YKimura K(2022)Quantaloidal Approach to Constraint SatisfactionElectronic Proceedings in Theoretical Computer Science10.4204/EPTCS.372.21372(289-305)Online publication date: 3-Nov-2022
https://doi.org/10.4204/EPTCS.372.21
Show More Cited By

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents