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The Complexity of Promise SAT on Non-Boolean Domains

Authors Alex Brandts, Marcin Wrochna , Stanislav Živný



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Author Details

Alex Brandts
  • Department of Computer Science, University of Oxford, UK
Marcin Wrochna
  • Department of Computer Science, University of Oxford, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Acknowledgements

We would like to thank the anonymous reviewers for their comments.

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Alex Brandts, Marcin Wrochna, and Stanislav Živný. The Complexity of Promise SAT on Non-Boolean Domains. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 17:1-17:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.17

Abstract

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and Håstad [FOCS'14/SICOMP'17] proved a result known as "(2+ε)-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Constraint and logic programming
Keywords
  • promise constraint satisfaction
  • PCSP
  • polymorphisms
  • algebraic approach
  • label cover

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References

  1. Per Austrin, Amey Bhangale, and Aditya Potukuchi. Improved inapproximability of rainbow coloring. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'20), pages 1479-1495. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.90.
  2. Per Austrin, Venkatesan Guruswami, and Johan Håstad. (2+ε)-SAT Is NP-hard. SIAM Journal on Computing, 46(5):1554-1573, 2017. URL: https://doi.org/10.1137/15M1006507.
  3. Libor Barto, Jakub Bulín, Andrei A. Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. arXiv:1811.00970, 2019. Version 3, 21 June 2019. URL: http://arxiv.org/abs/1811.00970.
  4. Libor Barto, Andrei Krokhin, and Ross Willard. Polymorphisms, and how to use them. In Andrei Krokhin and Stanislav Živný, editors, Complexity and approximability of Constraint Satisfaction Problems, volume 7 of Dagstuhl Follow-Ups, pages 1-44. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/DFU.Vol7.15301.i.
  5. Joshua Brakensiek and Venkatesan Guruswami. New Hardness Results for Graph and Hypergraph Colorings. In Proceedings of the 31st Conference on Computational Complexity (CCC'16), volume 50 of LIPIcs, pages 14:1-14:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.14.
  6. Joshua Brakensiek and Venkatesan Guruswami. Promise Constraint Satisfaction: Structure Theory and a Symmetric Boolean Dichotomy. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'18), pages 1782-1801. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.117.
  7. Joshua Brakensiek and Venkatesan Guruswami. 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), pages 436-455. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.28.
  8. Joshua Brakensiek and Venkatesan Guruswami. Symmetric polymorphisms and efficient decidability of PCSPs. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'20), pages 297-304. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.18.
  9. Alex Brandts, Marcin Wrochna, and Stanislav Živný. The complexity of promise SAT on non-Boolean domains. CoRR, abs/1911.09065, 2019. URL: http://arxiv.org/abs/1911.09065.
  10. Jakub Bulín, Andrei A. Krokhin, and Jakub Opršal. Algebraic approach to promise constraint satisfaction. In Proceedings of the 51st Annual ACM SIGACT Symposium on the Theory of Computing (STOC'19), pages 602-613. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316300.
  11. Hubie Chen and Martin Grohe. Constraint satisfaction with succinctly specified relations. J. Comput. Syst. Sci., 76(8):847-860, 2010. URL: https://doi.org/10.1016/j.jcss.2010.04.003.
  12. Stephen Cook. The complexity of theorem proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing (STOC'71), pages 151-158, 1971. URL: https://doi.org/10.1145/800157.805047.
  13. Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput., 34(5):1129-1146, 2005. URL: https://doi.org/10.1137/S0097539704443057.
  14. Irit Dinur, Oded Regev, and Clifford D. Smyth. The hardness of 3-uniform hypergraph coloring. Combinatorica, 25(5):519-535, 2005. URL: https://doi.org/10.1007/s00493-005-0032-4.
  15. Miron Ficak, Marcin Kozik, Miroslav Olšák, and Szymon Stankiewicz. Dichotomy for Symmetric Boolean PCSPs. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP'19), volume 132, pages 57:1-57:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.57.
  16. M. R. Garey and David S. Johnson. The complexity of near-optimal graph coloring. J. ACM, 23(1):43-49, 1976. URL: https://doi.org/10.1145/321921.321926.
  17. Àngel J. Gil, Miki Hermann, Gernot Salzer, and Bruno Zanuttini. Efficient algorithms for description problems over finite totally ordered domains. SIAM J. Comput., 38(3):922-945, 2008. URL: https://doi.org/10.1137/050635900.
  18. Venkatesan Guruswami and Sai Sandeep. Rainbow coloring hardness via low sensitivity polymorphisms. SIAM J. Discrete Math., 34(1):520-537, 2020. URL: https://doi.org/10.1137/19M127731X.
  19. Subhash Khot. Hardness results for coloring 3-colorable 3-uniform hypergraphs. In Proc. 43rd Symposium on Foundations of Computer Science (FOCS 2002), pages 23-32. IEEE Computer Society, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181879.
  20. Andrei Krokhin and Jakub Opršal. The complexity of 3-colouring H-colourable graphs. In Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS'19), pages 1227-1239. IEEE, 2019. URL: https://doi.org/10.1109/FOCS.2019.00076.
  21. Melven Krom. The decision problem for a class of first-order formulas in which all disjunctions are binary. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 13:15-20, 1967. URL: https://doi.org/10.1002/malq.19670130104.
  22. Leonid Levin. Universal search problems (in Russian). Problems of Information Transmission (in Russian), 9(3):115-116, 1973. URL: http://www.mathnet.ru/links/84e4c96b64cc22a33a4bdae3d4815887/ppi914.pdf.
  23. Christos H. Papadimitriou. On selecting a satisfying truth assignment. In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science (FOCS'91), pages 163-169. IEEE Computer Society, 1991. URL: https://doi.org/10.1109/SFCS.1991.185365.
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