research-article

On the complexity of #CSP

Authors:

Martin E. Dyer,

David M. RicherbyAuthors Info & Claims

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

Pages 725 - 734

https://doi.org/10.1145/1806689.1806789

Published: 05 June 2010 Publication History

Abstract

Bulatov (2008) has given a dichotomy for the counting constraint satisfaction problem, #CSP. A problem from #CSP is characterized by a constraint language γ, which is a fixed, finite set of relations over a finite domain. An instance of the problem uses these relations to constrain the values taken by a finite set of variables. Bulatov showed that, for any fixed γ, the problem of counting the satisfying assignments of instances of any problem from #CSP is either in polynomial time (FP) or #P-complete, according on the structure of the constraint language γ. His proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field.

We give an elementary proof of Bulatov's dichotomy, based on succinct representations, which we call frames, of a class of highly structured relations, which we call strongly rectangular. We show that these are precisely the relations that are invariant under a Mal'tsev polymorphism. En route, we give a simplification of a decision algorithm for strongly rectangular constraint languages due to Bulatov and Dalmau (2006). Out proof uses no universal algebra, except for the straightforward concept of the Mal'tsev polymorphism and is accessible to readers with little background in #CSP.

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Cited By

Cai JMaran ASaha BServedio R(2023)The Complexity of Counting Planar Graph Homomorphisms of Domain Size 3Proceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585173(1285-1297)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585173
Liu Y(2023)Exponential Time Complexity of the Complex Weighted Boolean #CSPComputing and Combinatorics10.1007/978-3-031-49190-0_6(83-96)Online publication date: 15-Dec-2023
https://dl.acm.org/doi/10.1007/978-3-031-49190-0_6
Roth MWellnitz PChawla S(2020)Counting and finding homomorphisms is universal for parameterized complexity theoryProceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3381089.3381222(2161-2180)Online publication date: 5-Jan-2020
https://dl.acm.org/doi/10.5555/3381089.3381222
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Index Terms

On the complexity of #CSP
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

June 2010

812 pages

ISBN:9781450300506

DOI:10.1145/1806689

General Chair:
Michael Mitzenmacher
Harvard
,
Program Chair:
Leonard J. Schulman
Caltech

Copyright © 2010 ACM.

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Published: 05 June 2010

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STOC'10: Symposium on Theory of Computing

June 5 - 8, 2010

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Cited By

Cai JMaran ASaha BServedio R(2023)The Complexity of Counting Planar Graph Homomorphisms of Domain Size 3Proceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585173(1285-1297)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585173
Liu Y(2023)Exponential Time Complexity of the Complex Weighted Boolean #CSPComputing and Combinatorics10.1007/978-3-031-49190-0_6(83-96)Online publication date: 15-Dec-2023
https://dl.acm.org/doi/10.1007/978-3-031-49190-0_6
Roth MWellnitz PChawla S(2020)Counting and finding homomorphisms is universal for parameterized complexity theoryProceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3381089.3381222(2161-2180)Online publication date: 5-Jan-2020
https://dl.acm.org/doi/10.5555/3381089.3381222
Cai JLu PXia M(2020)Dichotomy for Holant∗ Problems on the Boolean DomainTheory of Computing Systems10.1007/s00224-020-09983-8Online publication date: 22-Jun-2020
https://doi.org/10.1007/s00224-020-09983-8
Cai JKowalczyk MWilliams T(2019)Gadgets and Anti-Gadgets Leading to a Complexity DichotomyACM Transactions on Computation Theory10.1145/330527211:2(1-26)Online publication date: 17-Feb-2019
https://dl.acm.org/doi/10.1145/3305272
Cai JChen X(2019)A decidable dichotomy theorem on directed graph homomorphisms with non-negative weightscomputational complexity10.1007/s00037-019-00184-5Online publication date: 22-Apr-2019
https://doi.org/10.1007/s00037-019-00184-5
Cai JLu PXia MCzumaj A(2018)Dichotomy for real holant problemsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175423(1802-1821)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175423
Cai JGuo HWilliams T(2016)The complexity of counting edge colorings and a dichotomy for some higher domain Holant problemsResearch in the Mathematical Sciences10.1186/s40687-016-0067-83:1Online publication date: 1-Sep-2016
https://doi.org/10.1186/s40687-016-0067-8
Huang SLu P(2016)A Dichotomy for Real Weighted Holant ProblemsComputational Complexity10.1007/s00037-015-0118-325:1(255-304)Online publication date: 1-Mar-2016
https://dl.acm.org/doi/10.1007/s00037-015-0118-3
Cai JGuo HWilliams T(2014)The Complexity of Counting Edge Colorings and a Dichotomy for Some Higher Domain Holant ProblemsProceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science10.1109/FOCS.2014.70(601-610)Online publication date: 18-Oct-2014
https://dl.acm.org/doi/10.1109/FOCS.2014.70
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