research-article

Schaefer's Theorem for Graphs

Authors:

Manuel Bodirsky,

Michael PinskerAuthors Info & Claims

Journal of the ACM (JACM), Volume 62, Issue 3

Article No.: 19, Pages 1 - 52

https://doi.org/10.1145/2764899

Published: 30 June 2015 Publication History

Abstract

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete.

We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction Φ of statements (“constraints”) about these variables in the language of graphs, where each statement is taken from a fixed finite set Ψ of allowed quantifier-free first-order formulas; the question is whether Φ is satisfiable in a graph.

We prove that either Ψ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universal-algebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are first-order definable in the countably infinite random graph. Our method for classifying the computational complexity of those CSPs is based on a Ramsey-theoretic analysis of functions acting on the random graph, and we develop general tools suitable for such an analysis which are of independent mathematical interest.

References

[1]

F. G. Abramson and L. Harrington. 1978. Models without indiscernibles. J. Symb. Logic 43, 3, 572--600.

[2]

M. Bodirsky. 2012. Complexity classification in infinite-domain constraint satisfaction. Mémoire d'habilitation à diriger des recherches, Université Diderot -- Paris 7. arXiv:1201.0856.

[3]

M. Bodirsky, H. Chen, J. Kára, and T. von Oertzen. 2009. Maximal infinite-valued constraint languages. Theoret. Comput. Sci. (TCS) 410, 1684--1693. (A preliminary version appeared in Proceedings of ICALP'07.)

Digital Library

[4]

M. Bodirsky, H. Chen, and M. Pinsker. 2010. The reducts of equality up to primitive positive interdefinability. J. Symb. Logic 75, 4, 1249--1292.

[5]

M. Bodirsky, P. Jonsson, and T. von Oertzen. 2011. Horn versus full first-order: A complexity dichotomy for algebraic constraint satisfaction problems. J. Logic Comput. 22, 3, 643--660.

Digital Library

[6]

M. Bodirsky and J. Kára. 2008. The complexity of equality constraint languages. Theory of Comput. Syst. 3, 2, 136--158. (A conference version appeared in Proceedings of Computer Science Russia (CSR'06).)

[7]

M. Bodirsky and J. Kára. 2009. The complexity of temporal constraint satisfaction problems. J. ACM 57, 2, 1--41. (An extended abstract appeared in the Proceedings of the Symposium on Theory of Computing (STOC'08).)

Digital Library

[8]

M. Bodirsky and J. Kára. 2010. A fast algorithm and Datalog inexpressibility for temporal reasoning. ACM Trans. Computat. Logic 11, 3.

Digital Library

[9]

M. Bodirsky and J. Nešetřril. 2006. Constraint satisfaction with countable homogeneous templates. J. Logic Computat. 16, 3, 359--373.

Digital Library

[10]

M. Bodirsky and M. Pinsker. 2011a. Reducts of Ramsey structures. AMS Contemp. Math. 558 (Model Theoretic Methods in Finite Combinatorics), 489--519.

[11]

M. Bodirsky and M. Pinsker. 2011b. Schaefer's theorem for graphs. In Proceedings of the Annual Symposium on Theory of Computing (STOC). 655--664.

Digital Library

[12]

M. Bodirsky and M. Pinsker. 2014. Minimal functions on the random graph. Isr. J. Math. 200, 1, 251--296.

[13]

M. Bodirsky and M. Pinsker. 2015. Topological Birkhoff. Trans. Amer. Math. Soc. 367, 2527--2549.

[14]

M. Bodirsky, M. Pinsker, and A. Pongrácz. 2013a. The 42 reducts of the random ordered graph. Proc. LMS. To appear. (Preprint arXiv:1309.2165.)

[15]

M. Bodirsky, M. Pinsker, and T. Tsankov. 2013b. Decidability of definability. J. Symb. Logic 78, 4, 1036--1054. (A conference version appeared in Proceedings of LICS 2011, pages 321--328.)

Digital Library

[16]

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

Digital Library

[17]

P. J. Cameron. 1997. The random graph. Algor. Combinat. 14, 333--351.

[18]

P. J. Cameron. 2001. The random graph revisited. In Proceedings of the European Congress of Mathematics, vol. 201, Birkhäuser, 267--274.

[19]

H. Chen. 2006. A rendezvous of logic, complexity, and algebra. SIGACT News 37, 4, 85--114.

Digital Library

[20]

D. A. Cohen, P. Jeavons, P. Jonsson, and M. Koubarakis. 2000. Building tractable disjunctive constraints. J. ACM 47, 5, 826--853.

Digital Library

[21]

I. Duentsch. 2005. Relation algebras and their application in temporal and spatial reasoning. Artifi. Intell. Rev. 23, 315--357.

Digital Library

[22]

G. Exoo. 1989. A lower bound for R(5,5). J. Graph Theory 13, 97--98.

[23]

M. Garey and D. Johnson. 1978. A Guide to NP-Completeness. CSLI Press, Stanford.

[24]

M. Goldstern and M. Pinsker. 2008. A survey of clones on infinite sets. Algebra Univ. 59, 365--403.

[25]

W. Hodges. 1997. A Shorter Model Theory. Cambridge University Press, Cambridge.

Digital Library

[26]

P. Jeavons, D. Cohen, and M. Gyssens. 1997. Closure properties of constraints. J. ACM 44, 4, 527--548.

Digital Library

[27]

J. Nešetřril. 1995. Ramsey theory. Handbook of Combinatorics, 1331--1403.

Digital Library

[28]

J. Nešetřril and V. Rödl. 1983. Ramsey classes of set systems. J. Combinat. Theory, Series A 34, 2, 183--201.

[29]

T. J. Schaefer. 1978. The complexity of satisfiability problems. In Proceedings of the Symposium on Theory of Computing (STOC). 216--226.

Digital Library

[30]

Á. Szendrei. 1986. Clones in Universal Algebra. Séminaire de Mathématiques Supérieures. Les Presses de l'Université de Montréal.

[31]

S. Thomas. 1991. Reducts of the random graph. J. Symb. Logic 56, 1, 176--181.

Digital Library

[32]

S. Thomas. 1996. Reducts of random hypergraphs. Ann. Pure Appl. Logic 80, 2, 165--193.

[33]

M. Westphal and S. Wölfl. 2009. Qualitative CSP, finite CSP, and SAT: Comparing methods for qualitative constraint-based reasoning. In Proceedings of the 21th International Joint Conference on Artificial Intelligence (IJCAI). 628--633.

Digital Library

Cited By

Mottet APinsker M(2024)Smooth approximations: An algebraic approach to CSPs over finitely bounded homogeneous structuresJournal of the ACM10.1145/368920771:5(1-47)Online publication date: 17-Aug-2024
https://dl.acm.org/doi/10.1145/3689207
Barto LButti SKazda AViola CŽivný SSobocinski PLago UEsparza J(2024)Algebraic Approach to ApproximationProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662076(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662076
Mottet ANagy TPinsker MWrona M(2024)Collapsing the Bounded Width Hierarchy for Infinite-Domain Constraint Satisfaction Problems: When Symmetries Are EnoughSIAM Journal on Computing10.1137/22M153893453:6(1709-1745)Online publication date: 11-Dec-2024
https://doi.org/10.1137/22M1538934
Show More Cited By

Index Terms

Schaefer's Theorem for Graphs
1. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Schaefer's theorem for graphs
STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial ...
A Complexity Dichotomy in Spatial Reasoning via Ramsey Theory
Constraint satisfaction problems (CSPs) for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to obtain polynomial-...
Decidability of Definability
LICS '11: Proceedings of the 2011 IEEE 26th Annual Symposium on Logic in Computer Science

For a fixed infinite structure $\Gamma$ with finite signature tau, we study the following computational problem: input are quantifier-free first-order tau-formulas phi_0,phi_1,...,phi_n that define relations R_0,R_1,\dots,R_n over Gamma. The question is ...

Comments

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 62, Issue 3

June 2015

263 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2799630

Editor:
Victor Vianu
University of California, San Diego

Issue’s Table of Contents

Copyright © 2015 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: 30 June 2015

Accepted: 01 April 2015

Revised: 01 April 2013

Received: 01 October 2011

Published in JACM Volume 62, Issue 3

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

European Research Council under the European Community's Seventh Framework Programme
APART fellowship of the Austrian Academy of Sciences as well as through project I836-N23 of the Austrian Science Fund (FWF)

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

25
Total Citations
View Citations
413
Total Downloads

Downloads (Last 12 months)19
Downloads (Last 6 weeks)0

Reflects downloads up to 13 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Mottet APinsker M(2024)Smooth approximations: An algebraic approach to CSPs over finitely bounded homogeneous structuresJournal of the ACM10.1145/368920771:5(1-47)Online publication date: 17-Aug-2024
https://dl.acm.org/doi/10.1145/3689207
Barto LButti SKazda AViola CŽivný SSobocinski PLago UEsparza J(2024)Algebraic Approach to ApproximationProceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3661814.3662076(1-14)Online publication date: 8-Jul-2024
https://dl.acm.org/doi/10.1145/3661814.3662076
Mottet ANagy TPinsker MWrona M(2024)Collapsing the Bounded Width Hierarchy for Infinite-Domain Constraint Satisfaction Problems: When Symmetries Are EnoughSIAM Journal on Computing10.1137/22M153893453:6(1709-1745)Online publication date: 11-Dec-2024
https://doi.org/10.1137/22M1538934
Bodirsky MJonsson PMartin BMottet ASemanišinová Ž(2024)Complexity Classification Transfer for CSPs via Algebraic ProductsSIAM Journal on Computing10.1137/22M153430453:5(1293-1353)Online publication date: 12-Sep-2024
https://doi.org/10.1137/22M1534304
BODOR B(2023)CLASSIFICATION OF -CATEGORICAL MONADICALLY STABLE STRUCTURESThe Journal of Symbolic Logic10.1017/jsl.2023.66(1-36)Online publication date: 19-Sep-2023
https://doi.org/10.1017/jsl.2023.66
Gillibert PJonušas JKompatscher MMottet APinsker M(2022)When Symmetries Are Not Enough: A Hierarchy of Hard Constraint Satisfaction ProblemsSIAM Journal on Computing10.1137/20M138347151:2(175-213)Online publication date: 9-Mar-2022
https://doi.org/10.1137/20M1383471
Pinsker M(2022)Current Challenges in Infinite-Domain Constraint Satisfaction: Dilemmas of the Infinite Sheep2022 IEEE 52nd International Symposium on Multiple-Valued Logic (ISMVL)10.1109/ISMVL52857.2022.00019(80-87)Online publication date: May-2022
https://doi.org/10.1109/ISMVL52857.2022.00019
Gasarch WLaskowski MZhu S(2022), and Their CSP’sTheory and Applications of Models of Computation10.1007/978-3-031-20350-3_14(155-175)Online publication date: 16-Sep-2022
https://dl.acm.org/doi/10.1007/978-3-031-20350-3_14
Bodirsky MBodor B(2021)Permutation groups with small orbit growthJournal of Group Theory10.1515/jgth-2018-022024:4(643-709)Online publication date: 20-Jan-2021
https://doi.org/10.1515/jgth-2018-0220
Viola CŽivný S(2021)The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite DomainsACM Transactions on Algorithms10.1145/345804117:3(1-23)Online publication date: 15-Jul-2021
https://dl.acm.org/doi/10.1145/3458041
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