research-article

The Complexity of Promise SAT on Non-Boolean Domains

Authors:

Marcin Wrochna,

Stanislav ŽivnýAuthors Info & Claims

ACM Transactions on Computation Theory (TOCT), Volume 13, Issue 4

Article No.: 26, Pages 1 - 20

https://doi.org/10.1145/3470867

Published: 01 September 2021 Publication History

Abstract

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin et al. [SICOMP’17] proved a result known as “(2+ɛ)-SAT is NP-hard.” They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e., some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus, we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains.

The hardness side is proved using the algebraic approach via a new general NP-hardness criterion on polymorphisms, which is based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient—the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation in problems such as PCSPs.

References

[1]

Sanjeev Arora and Boaz Barak. 2009. Computational Complexity: A Modern Approach. Cambridge University Press.

Digital Library

[2]

Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. 1998. Proof verification and the hardness of approximation problems. J. ACM 45, 3 (1998), 501–555.

Digital Library

[3]

Sanjeev Arora and Shmuel Safra. 1998. Probabilistic checking of proofs: A new characterization of NP. J. ACM 45, 1 (1998), 70–122.

Digital Library

[4]

Per Austrin, Amey Bhangale, and Aditya Potukuchi. 2020. Improved inapproximability of rainbow coloring. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). SIAM, 1479–1495.

[5]

Per Austrin, Venkatesan Guruswami, and Johan Håstad. 2017. (2+)-SAT Is NP-hard. SIAM J. Comput. 46, 5 (2017), 1554–1573.

Digital Library

[6]

Libor Barto, Jakub Bulín, Andrei A. Krokhin, and Jakub Opršal. 2019. Algebraic approach to promise constraint satisfaction. arXiv:1811.00970. Retrieved June 21, 2019 from https://arxiv.org/abs/1811.00970.

[7]

Libor Barto, Andrei Krokhin, and Ross Willard. 2017. Polymorphisms, and how to use them. In Complexity and Approximability of Constraint Satisfaction Problems, Andrei Krokhin and Stanislav Živný (Eds.). Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 1–44.

[8]

Joshua Brakensiek and Venkatesan Guruswami. 2016. New hardness results for graph and hypergraph colorings. In Proceedings of the 31st Conference on Computational Complexity (CCC’ 16) (LIPIcs), Vol. 50. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 14:1–14:27.

[9]

Joshua Brakensiek and Venkatesan Guruswami. 2018. Promise constraint satisfaction: Structure theory and a symmetric boolean dichotomy. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’18). SIAM, 1782–1801.

[10]

Joshua Brakensiek and Venkatesan Guruswami. 2019. An algorithmic blend of LPs and ring equations for promise CSPs. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’19). SIAM, 436–455.

[11]

Joshua Brakensiek and Venkatesan Guruswami. 2020. Symmetric polymorphisms and efficient decidability of PCSPs. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’20). SIAM, 297–304.

[12]

Alex Brandts, Marcin Wrochna, and Stanislav Živný. 2020. The complexity of promise SAT on non-Boolean domains. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP’20), Vol. 168. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 17:1–17:13.

[13]

Jakub Bulín, Andrei A. Krokhin, and Jakub Opršal. 2019. Algebraic approach to promise constraint satisfaction. In Proceedings of the 51st Annual ACM SIGACT Symposium on the Theory of Computing (STOC’19). ACM, 602–613.

Digital Library

[14]

Hubie Chen and Martin Grohe. 2010. Constraint satisfaction with succinctly specified relations. J. Comput. Syst. Sci. 76, 8 (2010), 847–860.

Digital Library

[15]

Stephen Cook. 1971. The complexity of theorem proving procedures. In Proceedings of the 3rd Annual ACM Symposium on Theory of Computing (STOC’71). 151–158.

Digital Library

[16]

Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. 2005. A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput. 34, 5 (2005), 1129–1146.

Digital Library

[17]

Irit Dinur, Oded Regev, and Clifford D. Smyth. 2005. The hardness of 3-Uniform hypergraph coloring. Combinatorica 25, 5 (2005), 519–535.

Digital Library

[18]

Miron Ficak, Marcin Kozik, Miroslav Olšák, and Szymon Stankiewicz. 2019. Dichotomy for symmetric boolean PCSPs. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP’19), Vol. 132. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 57:1–57:12.

[19]

M. R. Garey and David S. Johnson. 1976. The complexity of near-optimal graph coloring. J. ACM 23, 1 (1976), 43–49.

Digital Library

[20]

Àngel J. Gil, Miki Hermann, Gernot Salzer, and Bruno Zanuttini. 2008. Efficient algorithms for description problems over finite totally ordered domains. SIAM J. Comput. 38, 3 (2008), 922–945.

Digital Library

[21]

Venkatesan Guruswami and Euiwoong Lee. 2018. Strong inapproximability results on balanced rainbow-colorable hypergraphs. Combinatorica 38, 3 (2018), 547–599.

Digital Library

[22]

Venkatesan Guruswami and Sai Sandeep. 2020. Rainbow coloring hardness via low sensitivity polymorphisms. SIAM J. Discrete Math. 34, 1 (2020), 520–537.

[23]

Subhash Khot. 2002. Hardness results for coloring 3-Colorable 3-Uniform hypergraphs. In Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS’02). IEEE Computer Society, 23–32.

Digital Library

[24]

Andrei Krokhin and Jakub Opršal. 2019. The complexity of 3-colouring -colourable graphs. In Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS’19). IEEE, 1227–1239.

[25]

Melven Krom. 1967. The decision problem for a class of first-order formulas in which all disjunctions are binary. Z. Math. Logik Grund. Math. 13 (1967), 15–20.

[26]

Leonid Levin. 1973. Universal search problems (in Russian). Probl. Inf. Transm. (in Russian) 9, 3 (1973), 115–116. http://www.mathnet.ru/links/84e4c96b64cc22a33a4bdae3d4815887/ppi914.pdf

[27]

Christos H. Papadimitriou. 1991. On selecting a satisfying truth assignment. In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science (FOCS’91). IEEE Computer Society, 163–169.

Digital Library

[28]

Ran Raz. 1998. A parallel repetition theorem. SIAM J. Comput. 27, 3 (1998), 763–803.

Digital Library

Cited By

Nakajima TŽivný S(2024)On the complexity of symmetric vs. functional PCSPsACM Transactions on Algorithms10.1145/3673655Online publication date: 18-Jun-2024
https://dl.acm.org/doi/10.1145/3673655
Nakajima TŽivný S(2023)Linearly Ordered Colourings of HypergraphsACM Transactions on Computation Theory10.1145/357090914:3-4(1-19)Online publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1145/3570909
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
Show More Cited By

Index Terms

The Complexity of Promise SAT on Non-Boolean Domains
1. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness
  2. Logic
    1. Constraint and logic programming

Recommendations

1-in-3 vs. Not-All-Equal: Dichotomy of a broken promise
LICS '24: Proceedings of the 39th Annual ACM/IEEE Symposium on Logic in Computer Science

The 1-in-3 and Not-All-Eqal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Eqal problem can be solved in polynomial time. In the present work, we investigate this ...
(2 + epsilon)-Sat Is NP-Hard
FOCS '14: Proceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science

We prove the following hardness result for anatural promise variant of the classical CNF-satisfiabilityproblem: Given a CNF-formula where each clause has widthw and the guarantee that there exists an assignment satisfyingat least g = [w/2]--1 literals ...
The Complexity of Boolean Surjective General-Valued CSPs

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q ∪ {∞ })-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from D = {0,1}, and an optimal ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Computation Theory

ACM Transactions on Computation Theory Volume 13, Issue 4

December 2021

198 pages

ISSN:1942-3454

EISSN:1942-3462

DOI:10.1145/3481683

Editor:
Ryan O'Donnell
Carnegie Mellon University, USA

Issue’s Table of Contents

Copyright © 2021 Association for Computing Machinery.

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 September 2021

Accepted: 01 May 2021

Received: 01 February 2021

Published in TOCT Volume 13, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Refereed

Funding Sources

Royal Society Enhancement Award
NSERC PGS Doctoral Award
Royal Society University Research Fellowship
European Research Council (ERC) under the European Union’s Horizon 2020

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
88
Total Downloads

Downloads (Last 12 months)22
Downloads (Last 6 weeks)0

Reflects downloads up to 27 Jul 2024

Other Metrics

View Author Metrics

Citations

Cited By

Nakajima TŽivný S(2024)On the complexity of symmetric vs. functional PCSPsACM Transactions on Algorithms10.1145/3673655Online publication date: 18-Jun-2024
https://dl.acm.org/doi/10.1145/3673655
Nakajima TŽivný S(2023)Linearly Ordered Colourings of HypergraphsACM Transactions on Computation Theory10.1145/357090914:3-4(1-19)Online publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1145/3570909
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
Krokhin AOpršal JWrochna MŽivný S(2023)Topology and Adjunction in Promise Constraint SatisfactionSIAM Journal on Computing10.1137/20M137822352:1(38-79)Online publication date: 14-Feb-2023
https://doi.org/10.1137/20M1378223
Nakajima TŽivný S(2023)Boolean symmetric vs. functional PCSP dichotomy2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)10.1109/LICS56636.2023.10175746(1-12)Online publication date: 26-Jun-2023
https://doi.org/10.1109/LICS56636.2023.10175746
Fu ZWang HLiu JZhou JXu DPi Y(2022)The Transition Phenomenon of (1,0)-d-Regular (k, s)-SATElectronics10.3390/electronics1115247511:15(2475)Online publication date: 8-Aug-2022
https://doi.org/10.3390/electronics11152475
Krokhin AOpršal J(2022)An invitation to the promise constraint satisfaction problemACM SIGLOG News10.1145/3559736.35597409:3(30-59)Online publication date: 25-Aug-2022
https://dl.acm.org/doi/10.1145/3559736.3559740
Brandts AŽivný S(2022)Beyond PCSP(1-in-3,NAE)Information and Computation10.1016/j.ic.2022.104954289:PAOnline publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1016/j.ic.2022.104954

View Options

Get Access

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.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents