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The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

Published:15 December 2021Publication History
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Abstract

We study the fine-grained complexity of NP-complete satisfiability (SAT) problems and constraint satisfaction problems (CSPs) in the context of the strong exponential-time hypothesis(SETH), showing non-trivial lower and upper bounds on the running time. Here, by a non-trivial lower bound for a problem SAT (Γ) (respectively CSP (Γ)) with constraint language Γ, we mean a value c0 > 1 such that the problem cannot be solved in time O(cn) for any c <c0 unless SETH is false, while a non-trivial upper bound is simply an algorithm for the problem running in time O(cn) for some c< 2. Such lower bounds have proven extremely elusive, and except for cases where c0=2 effectively no such previous bound was known. We achieve this by employing an algebraic framework, studying constraint languages Γ in terms of their algebraic properties. We uncover a powerful algebraic framework where a mild restriction on the allowed constraints offers a concise algebraic characterization. On the relational side we restrict ourselves to Boolean languages closed under variable negation and partial assignment, called sign-symmetric languages. On the algebraic side this results in a description via partial operations arising from system of identities, with a close connection to operations resulting in tractable CSPs, such as near unanimity operations and edge operations. Using this connection we construct improved algorithms for several interesting classes of sign-symmetric languages, and prove explicit lower bounds under SETH. Thus, we find the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits a non-trivial lower bound under SETH. This suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem: an NP-complete SAT problem admits an improved algorithm if and only if it admits a non-trivial partial invariant of the above form.

REFERENCES

  1. [1] Alon Noga, Shpilka Amir, and Umans Christopher. 2013. On sunflowers and matrix multiplication. Computational Complexity 22, 2 (2013), 219243. Google ScholarGoogle ScholarCross RefCross Ref
  2. [2] Alon Noga, Yuster Raphael, and Zwick Uri. 1997. Finding and counting given length cycles. Algorithmica 17, 3 (1997), 209223. Google ScholarGoogle ScholarCross RefCross Ref
  3. [3] Alweiss Ryan, Lovett Shachar, Wu Kewen, and Zhang Jiapeng. 2020. Improved bounds for the sunflower lemma. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC-2020). ACM, 624630.Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. [4] Bansal Nikhil, Garg Shashwat, Nederlof Jesper, and Vyas Nikhil. 2017. Faster space-efficient algorithms for subset sum and k-sum. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC-2017). ACM, 198209.Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. [5] Barto L., Krokhin A., and Willard R.. 2017. Polymorphisms, and how to use them. In The Constraint Satisfaction Problem: Complexity and Approximability, Krokhin Andrei and Zivny Stanislav (Eds.). Dagstuhl Follow-Ups, Vol. 7. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 144. Google ScholarGoogle ScholarCross RefCross Ref
  6. [6] Barto L., Opršal J., and Pinsker M.. 2018. The wonderland of reflections. Israel Journal of Mathematics 223, 1 (01 Feb 2018), 363398. Google ScholarGoogle ScholarCross RefCross Ref
  7. [7] Berman J., Idziak P., Markovic P., McKenzie R., Valeriote M., and Willard R.. 2010. Varieties with few subalgebras of powers.Trans. Amer. Math. Soc. 362, 3 (2010), 14451473. Google ScholarGoogle ScholarCross RefCross Ref
  8. [8] Björklund Andreas, Husfeldt Thore, and Koivisto Mikko. 2009. Set partitioning via inclusion-exclusion. SIAM J. Comput. 39, 2 (2009), 546563. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. [9] Brakensiek J. and Guruswami V.. 2019. Bridging between 0/1 and linear programming via random walks. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC-2019). ACM, New York, NY.568577.Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. [10] Bringmann Karl, Fischer Nick, and Künnemann Marvin. 2019. A fine-grained analogue of Schaefer’s theorem in P: Dichotomy of exists2303k-forall-quantified first-order graph properties. In Proceedings of the 34th Computational Complexity Conference (CCC-2019) (LIPIcs), Vol. 137. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 31:1–31:27.Google ScholarGoogle Scholar
  11. [11] Bulatov A.. 2017. A dichotomy theorem for nonuniform CSPs. In Proceedings of the 58th Annual Symposium on Foundations of Computer Science (FOCS-2017). IEEE Computer Society.Google ScholarGoogle ScholarCross RefCross Ref
  12. [12] Bulatov A. and Dalmau V.. 2006. A simple algorithm for Mal’tsev constraints. SIAM Journal On Computing 36, 1 (2006), 1627.Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. [13] Bulatov Andrei A.. 2018. Constraint satisfaction problems: Complexity and algorithms. ACM SIGLOG News 5, 4 (Nov. 2018), 424. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. [14] Calabro Chris, Impagliazzo Russell, and Paturi Ramamohan. 2009. The complexity of satisfiability of small depth circuits. In Parameterized and Exact Computation, 4th International Workshop (IWPEC 2009). 7585. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. [15] Calabro C., Impagliazzo R., and Paturi R.. 2013. On the exact complexity of evaluating quantified k-CNF. Algorithmica 65, 4 (01 Apr 2013), 817827. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. [16] Chen H., Jansen B. M. P., and Pieterse A.. 2020. Best-case and worst-case sparsifiability of boolean CSPs. Algorithmica 82, 8 (2020), 22002242. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. [17] Cormen T. H., Leiserson C. E., Rivest R. L., and Stein C.. 2009. Introduction to Algorithms, Third Edition (3rd ed.). The MIT Press.Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. [18] Couceiro M., Haddad L., Lagerkvist V., and Roy B.. 2017. On the interval of boolean strong partial clones containing only projections as total operations. In Proceedings of the 47th International Symposium on Multiple-Valued Logic (ISMVL-2017). IEEE Computer Society, 8893.Google ScholarGoogle ScholarCross RefCross Ref
  19. [19] Cygan Marek, Dell Holger, Lokshtanov Daniel, Marx Dániel, Nederlof Jesper, Okamoto Yoshio, Paturi Ramamohan, Saurabh Saket, and Wahlström Magnus. 2016. On problems as hard as CNF-SAT. ACM Transactions on Algorithms 12, 3 (2016), 41:1–41:24. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. [20] Cygan Marek, Fomin Fedor V., Kowalik Lukasz, Lokshtanov Daniel, Marx Dániel, Pilipczuk Marcin, Pilipczuk Michal, and Saurabh Saket. 2015. Parameterized Algorithms. Springer.Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. [21] Dantsin Evgeny, Goerdt Andreas, Hirsch Edward A., Kannan Ravi, Kleinberg Jon M., Papadimitriou Christos H., Raghavan Prabhakar, and Schöning Uwe. 2002. A deterministic algorithm for k-SAT based on local search. Theoretical Computer Science 289, 1 (2002), 6983. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. [22] Erdős P. and Rado R.. 1960. Intersection theorems for systems of sets. Journal of the London Mathematical Society s1-35, 1 (1960), 8590. Google ScholarGoogle ScholarCross RefCross Ref
  23. [23] Fomin F. V., Lokshtanov D., Saurabh S., and Zehavi M.. 2019. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press. 2018030108 https://books.google.se/books?id=ZsMouQEACAAJ.Google ScholarGoogle Scholar
  24. [24] Fomin Fedor V., Gaspers Serge, Lokshtanov Daniel, and Saurabh Saket. 2016. Exact algorithms via monotone local search. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2016). 764775. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. [25] Geiger D.. 1968. Closed systems of functions and predicates. Pacific J. Math. 27, 1 (1968), 95100.Google ScholarGoogle ScholarCross RefCross Ref
  26. [26] Hansen Thomas Dueholm, Kaplan Haim, Zamir Or, and Zwick Uri. 2019. Faster k-SAT algorithms using biased-PPSZ. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC-2019). ACM, 578589.Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. [27] Harnik D. and Naor M.. 2010. On the compressibility of NP instances and cryptographic applications. SIAM J. Comput. 39, 5 (2010), 16671713. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. [28] Hertli T.. 2014. 3-SAT faster and simpler - unique-SAT bounds for PPSZ hold in general. SIAM J. Comput. 43, 2 (2014), 718729.Google ScholarGoogle ScholarCross RefCross Ref
  29. [29] Hertli Timon. 2014. Breaking the PPSZ barrier for unique 3-SAT. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP-2014) (Lecture Notes in Computer Science), Vol. 8572. Springer, 600611.Google ScholarGoogle ScholarCross RefCross Ref
  30. [30] Horowitz E. and Sahni S.. 1974. Computing partitions with applications to the knapsack problem. J. ACM 21, 2 (April 1974), 277292.Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. [31] Idziak P., Marković P., McKenzie R., Valeriote M., and Willard R.. 2010. Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39, 7 (June 2010), 30233037.Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. [32] Impagliazzo R. and Paturi R.. 2001. On the complexity of k-SAT. J. Comput. System Sci. 62, 2 (2001), 367375.Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. [33] Impagliazzo R., Paturi R., and Zane F.. 2001. Which problems have strongly exponential complexity?J. Comput. System Sci. 63 (2001), 512530. Issue 4.Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. [34] Jansen B. M. P. and Pieterse A.. 2016. Optimal sparsification for some binary CSPs using low-degree polynomials. In Proceedings of the 41st International Symposium on Mathematical Foundations of Computer Science (MFCS-2016), Vol. 58. 71:1–71:14.Google ScholarGoogle Scholar
  35. [35] Jeavons P., Cohen D., and Gyssens M.. 1997. Closure properties of constraints. J. ACM 44, 4 (July 1997), 527548. Google ScholarGoogle ScholarDigital LibraryDigital Library
  36. [36] Jonsson P., Lagerkvist V., Nordh G., and Zanuttini B.. 2017. Strong partial clones and the time complexity of SAT problems. J. Comput. System Sci. 84 (2017), 5278.Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. [37] Jonsson P., Lagerkvist V., and Roy B.. 2021. Fine-grained time complexity of constraint satisfaction problems. ACM Transactions on Computation Theory 13, 1, Article 2 (2021), 32 pages.Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. [38] Künnemann Marvin and Marx Dániel. 2020. Finding small satisfying assignments faster than brute force: A fine-grained perspective into boolean constraint satisfaction. In Computational Complexity Conference (LIPIcs), Vol. 169. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 27:1–27:28.Google ScholarGoogle Scholar
  39. [39] Lagerkvist V.. 2016. Strong Partial Clones and the Complexity of Constraint Satisfaction Problems: Limitations and Applications. Ph.D. Dissertation. Linköping University, The Institute of Technology.Google ScholarGoogle Scholar
  40. [40] Lagerkvist V. and Roy B.. 2016. A preliminary investigation of satisfiability problems not harder than 1-in-3-SAT. In Proceedings of the 41st International Symposium on Mathematical Foundations of Computer Science (MFCS-2016). 64:1–64:14.Google ScholarGoogle Scholar
  41. [41] Lagerkvist V. and Wahlström M.. 2017. The power of primitive positive definitions with polynomially many variables. Journal of Logic and Computation 27, 5 (2017), 14651488. Google ScholarGoogle ScholarCross RefCross Ref
  42. [42] Lagerkvist V. and Wahlström M.. 2020. Sparsification of SAT and CSP problems via tractable extensions. ACM Transactions on Computation Theory 12, 2, Article 13 (2020), 29 pages.Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. [43] Lagerkvist V., Wahlström M., and Zanuttini B.. 2015. Bounded bases of strong partial clones. In Proceedings of the 45th International Symposium on Multiple-Valued Logic (ISMVL-2015). 189194.Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. [44] Gall François Le. 2014. Powers of tensors and fast matrix multiplication. In Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC-2014). 296303. Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. [45] Lincoln A., Williams V. Vassilevska, and Williams R.. 2018. Tight hardness for shortest cycles and paths in sparse graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2018). 12361252.Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. [46] Lincoln Andrea and Yedidia Adam. 2020. Faster random k-CNF satisfiability. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP-2020) (LIPIcs), Vol. 168. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 78:1–78:12.Google ScholarGoogle Scholar
  47. [47] Lokshtanov D., Marx D., and Saurabh S.. 2011. Known algorithms on graphs of bounded treewidth are probably optimal. In Proceedings of the Twenty-second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2011). 777789. http://dl.acm.org/citation.cfm?id=2133036.2133097.Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. [48] Lokshtanov Daniel, Paturi Ramamohan, Tamaki Suguru, Williams R. Ryan, and Yu Huacheng. 2017. Beating brute force for systems of polynomial equations over finite fields. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA-2017), Klein Philip N. (Ed.). 21902202. Google ScholarGoogle ScholarCross RefCross Ref
  49. [49] Meeks Kitty. 2016. Randomised enumeration of small witnesses using a decision oracle. In 11th International Symposium on Parameterized and Exact Computation (IPEC-2016). 22:1–22:12. Google ScholarGoogle ScholarCross RefCross Ref
  50. [50] Monien Burkhard and Speckenmeyer Ewald. 1985. Solving satisfiability in less than 2 steps. Discrete Applied Mathematics 10, 3 (1985), 287295.Google ScholarGoogle ScholarDigital LibraryDigital Library
  51. [51] Nisan Noam. 1991. CREW PRAMs and decision trees. SIAM Journal On Computing 20, 6 (1991), 9991007. Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. [52] O’Donnell Ryan. 2014. Analysis of Boolean Functions. Cambridge University Press. http://www.cambridge.org/de/academic/subjects/computer-science/algorithmics-complexity-computer-algebra-and-computational-g/analysis-boolean-functions.Google ScholarGoogle ScholarCross RefCross Ref
  53. [53] Paturi Ramamohan, Pudlák Pavel, Saks Michael E., and Zane Francis. 2005. An improved exponential-time algorithm for k-SAT. J. ACM 52, 3 (2005), 337364.Google ScholarGoogle ScholarDigital LibraryDigital Library
  54. [54] Romov B. A.. 1981. The algebras of partial functions and their invariants. Cybernetics 17, 2 (1981), 157167.Google ScholarGoogle Scholar
  55. [55] Schaefer T.. 1978. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory Of Computing (STOC-1978). ACM Press, 216226.Google ScholarGoogle ScholarDigital LibraryDigital Library
  56. [56] Scheder D. and Steinberger J. P.. 2017. PPSZ for general k-SAT - making Hertli’s analysis simpler and 3-SAT faster. In Proceedings of the 32nd Computational Complexity Conference (CCC-2017). 9:1–9:15. Google ScholarGoogle ScholarCross RefCross Ref
  57. [57] Scheder Dominik and Talebanfard Navid. 2020. Super strong ETH is true for PPSZ with small resolution width. In Proceedings of the 35th Computational Complexity Conference (CCC-2020) (LIPIcs), Vol. 169. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 3:1–3:12.Google ScholarGoogle Scholar
  58. [58] Schnoor H. and Schnoor I.. 2008. Partial polymorphisms and constraint satisfaction problems. In Complexity of Constraints, Creignou N., Kolaitis P. G., and Vollmer H. (Eds.). Lecture Notes in Computer Science, Vol. 5250. Springer Berlin, 229254.Google ScholarGoogle ScholarDigital LibraryDigital Library
  59. [59] Schöning U.. 1999. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS-1999). 410414.Google ScholarGoogle ScholarCross RefCross Ref
  60. [60] Vyas Nikhil and Williams R. Ryan. 2019. On super strong ETH. In Proceedings of the 22nd International Conference on Theory and Applications of Satisfiability Testing (SAT-2019) (Lecture Notes in Computer Science), Vol. 11628. Springer, 406423.Google ScholarGoogle ScholarCross RefCross Ref
  61. [61] Wahlström M.. 2007. Algorithms, Measures and Upper Bounds for Satisfiability and Related Problems. Ph.D. Dissertation. Linköping University, TCSLAB - Theoretical Computer Science Laboratory, The Institute of Technology.Google ScholarGoogle Scholar
  62. [62] Williams R.. 2005. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science 348, 2 (2005), 357365. Automata, Languages and Programming: Algorithms and Complexity (ICALP-A 2004).Google ScholarGoogle ScholarDigital LibraryDigital Library
  63. [63] Williams Virginia Vassilevska. 2012. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th Symposium on Theory of Computing Conference (STOC-2012). 887898. Google ScholarGoogle ScholarDigital LibraryDigital Library
  64. [64] Williams Virginia Vassilevska and Williams Ryan. 2013. Finding, minimizing, and counting weighted subgraphs. SIAM Journal On Computing 42, 3 (2013), 831854. Google ScholarGoogle ScholarDigital LibraryDigital Library
  65. [65] Xiao Mingyu and Nagamochi Hiroshi. 2017. Exact algorithms for maximum independent set. Information and Computation 255 (2017), 126146.Google ScholarGoogle ScholarCross RefCross Ref
  66. [66] Zhuk Dmitriy. 2020. A proof of the CSP dichotomy conjecture. Journal of the ACM 67, 5 (2020), 30:1–30:78.Google ScholarGoogle ScholarDigital LibraryDigital Library

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        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 14, Issue 1
        March 2022
        155 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3505197
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        Publication History

        • Published: 15 December 2021
        • Accepted: 1 September 2021
        • Revised: 1 June 2021
        • Received: 1 January 2021
        Published in toct Volume 14, Issue 1

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