On the Parallel Parameterized Complexity of MaxSAT Variants

Authors Max Bannach , Malte Skambath , Till Tantau

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Max Bannach
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany
Malte Skambath
  • Department of Computer Science, Universität Kiel, Germany
Till Tantau
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany

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Max Bannach, Malte Skambath, and Till Tantau. On the Parallel Parameterized Complexity of MaxSAT Variants. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In the maximum satisfiability problem (max-sat) we are given a propositional formula in conjunctive normal form and have to find an assignment that satisfies as many clauses as possible. We study the parallel parameterized complexity of various versions of max-sat and provide the first constant-time algorithms parameterized either by the solution size or by the allowed excess relative to some guarantee ("above guarantee" versions). For the dual parameterized version where the parameter is the number of clauses we are allowed to leave unsatisfied, we present the first parallel algorithm for max-2sat (known as almost-2sat). The difficulty in solving almost-2sat in parallel comes from the fact that the iterative compression method, originally developed to prove that the problem is fixed-parameter tractable at all, is inherently sequential. We observe that a graph flow whose value is a parameter can be computed in parallel and use this fact to develop a parallel algorithm for the vertex cover problem parameterized above the size of a given matching. Finally, we study the parallel complexity of max-sat parameterized by the vertex cover number, the treedepth, the feedback vertex set number, and the treewidth of the input’s incidence graph. While max-sat is fixed-parameter tractable for all of these parameters, we show that they allow different degrees of possible parallelization. For all four we develop dedicated parallel algorithms that are constructive, meaning that they output an optimal assignment - in contrast to results that can be obtained by parallel meta-theorems, which often only solve the decision version.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parallel computing models
  • Theory of computation → Fixed parameter tractability
  • max-sat
  • almost-sat
  • parallel algorithms
  • fixed-parameter tractability


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  1. Faisal N. Abu-Khzam and Karam Al Kontar. A Brief Survey of Fixed-Parameter Parallelism. Algorithms, 13(8):197, 2020. URL: https://doi.org/10.3390/a13080197.
  2. Noga Alon, Gregory Z. Gutin, Eun Jung Kim, Stefan Szeider, and Anders Yeo. Solving MAX-r-SAT Above a Tight Lower Bound. Algorithmica, 61(3):638-655, 2011. URL: https://doi.org/10.1007/s00453-010-9428-7.
  3. Max Bannach, Christoph Stockhusen, and Till Tantau. Fast Parallel Fixed-Parameter Algorithms via Color Coding. In 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, volume 43 of LIPIcs, pages 224-235. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.IPEC.2015.224.
  4. Max Bannach and Till Tantau. Parallel Multivariate Meta-Theorems. In 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, August 24-26, 2016, Aarhus, Denmark, volume 63 of LIPIcs, pages 4:1-4:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.IPEC.2016.4.
  5. Max Bannach and Till Tantau. Computing Kernels in Parallel: Lower and Upper Bounds. In 13th International Symposium on Parameterized and Exact Computation, IPEC 2018, August 20-24, 2018, Helsinki, Finland, volume 115 of LIPIcs, pages 13:1-13:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.IPEC.2018.13.
  6. David A. Mix Barrington, Neil Immerman, and Howard Straubing. On Uniformity within NC¹. Journal of Computer and System Sciences, 41(3):274-306, 1990. URL: https://doi.org/10.1016/0022-0000(90)90022-D.
  7. Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh, editors. Handbook of Satisfiability, Second Edition. IOS Press, 2021. Google Scholar
  8. Liming Cai, Jianer Chen, Rodney G. Downey, and Michael R. Fellows. Advice Classes of Parameterized Tractability. Annals of Pure and Applied Logic, 84(1):119-138, 1997. URL: https://doi.org/10.1016/S0168-0072(95)00020-8.
  9. Yijia Chen and Jörg Flum. Some Lower Bounds in Parameterized AC⁰. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, pages 27:1-27:14, 2016. URL: https://doi.org/10.4230/LIPIcs.MFCS.2016.27.
  10. Yijia Chen and Jörg Flum. Parameterized Parallel Computing and First-Order Logic. In Fields of Logic and Computation III - Essays Dedicated to Yuri Gurevich on the Occasion of His 80th Birthday, pages 57-78, 2020. URL: https://doi.org/10.1007/978-3-030-48006-6_5.
  11. Yijia Chen, Jörg Flum, and Xuangui Huang. Slicewise Definability in First-Order Logic with Bounded Quantifier Rank. In 26th EACSL Annual Conference on Computer Science Logic, CSL 2017, August 20-24, 2017, Stockholm, Sweden, volume 82 of LIPIcs, pages 19:1-19:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.CSL.2017.19.
  12. Bruno Courcelle. The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  13. Robert Crowston, Michael R. Fellows, Gregory Z. Gutin, Mark Jones, Frances A. Rosamond, Stéphan Thomassé, and Anders Yeo. Simultaneously Satisfying Linear Equations Over 𝔽₂: MaxLin2 and Max-r-Lin2 Parameterized Above Average. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2011, December 12-14, 2011, Mumbai, India, pages 229-240, 2011. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2011.229.
  14. Robert Crowston, Gregory Z. Gutin, Mark Jones, Venkatesh Raman, and Saket Saurabh. Parameterized Complexity of MaxSat Above Average. Theor. Comput. Sci., 511:77-84, 2013. URL: https://doi.org/10.1016/j.tcs.2013.01.005.
  15. Robert Crowston, Gregory Z. Gutin, Mark Jones, and Anders Yeo. A New Lower Bound on the Maximum Number of Satisfied Clauses in MaxSAT and Its Algorithmic Applications. Algorithmica, 64(1):56-68, 2012. URL: https://doi.org/10.1007/s00453-011-9550-1.
  16. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  17. Marek Cygan, Marcin Pilipczuk, Michal Pilipczuk, and Jakub Onufry Wojtaszczyk. On Multiway Cut Parameterized Above Lower Bounds. ACM Transactions on Computation Theory, 5(1):3:1-3:11, 2013. URL: https://doi.org/10.1145/2462896.2462899.
  18. Holger Dell, Eun Jung Kim, Michael Lampis, Valia Mitsou, and Tobias Mömke. Complexity and Approximability of Parameterized MAX-CSPs. Algorithmica, 79(1):230-250, 2017. URL: https://doi.org/10.1007/s00453-017-0310-8.
  19. Michael Elberfeld, Christoph Stockhusen, and Till Tantau. On the Space and Circuit Complexity of Parameterized Problems: Classes and Completeness. Algorithmica, 71(3):661-701, 2015. URL: https://doi.org/10.1007/s00453-014-9944-y.
  20. Bruno Escoffier and Vangelis Th. Paschos. Differential Approximation of MinSAT, MaxSAT and Related Problems. In Computational Science and Its Applications - ICCSA 2005, International Conference, Singapore, May 9-12, 2005, Proceedings, Part IV, pages 192-201, 2005. URL: https://doi.org/10.1007/11424925_22.
  21. Jörg Flum and Martin Grohe. Describing Parameterized Complexity Classes. Information and Computation, 187(2):291-319, 2003. URL: https://doi.org/10.1016/S0890-5401(03)00161-5.
  22. L. R. Ford and D. R. Fulkerson. Maximal Flow Through a Network. Canadian Journal of Mathematics, 8:399-404, 1956. URL: https://doi.org/10.4153/CJM-1956-045-5.
  23. M. R. Garey, David S. Johnson, and Larry J. Stockmeyer. Some Simplified NP-Complete Graph Problems. Theor. Comput. Sci., 1(3):237-267, 1976. URL: https://doi.org/10.1016/0304-3975(76)90059-1.
  24. Shivam Garg and Geevarghese Philip. Raising The Bar For Vertex Cover: Fixed-Parameter Tractability Above a Higher Guarantee. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1152-1166. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch80.
  25. Serge Gaspers and Stefan Szeider. Kernels for Global Constraints. In IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, July 16-22, 2011, pages 540-545, 2011. URL: https://doi.org/10.5591/978-1-57735-516-8/IJCAI11-098.
  26. Serge Gaspers and Stefan Szeider. Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning. Artif. Intell., 216:1-19, 2014. URL: https://doi.org/10.1016/j.artint.2014.06.006.
  27. Leslie M. Goldschlager, Ralph A. Shaw, and John Staples. The Maximum Flow Problem is Log Space Complete for P. Theoretical Computer Science, 21:105-111, 1982. URL: https://doi.org/10.1016/0304-3975(82)90092-5.
  28. Martin Grohe. The Structure of Tractable Constraint Satisfaction Problems. In Mathematical Foundations of Computer Science 2006, 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006, Proceedings, pages 58-72, 2006. URL: https://doi.org/10.1007/11821069_5.
  29. Gregory Z. Gutin, Mark Jones, Dominik Scheder, and Anders Yeo. A new Bound for 3-Satisfiable MaxSat and its Algorithmic Application. Inf. Comput., 231:117-124, 2013. URL: https://doi.org/10.1016/j.ic.2013.08.008.
  30. Dorit S. Hochbaum. Solving Integer Programs over Monotone Inequalities in three Variables: A Framework for Half Integrality and Good Approximations. European Journal of Operational Research, 140(2):291-321, 2002. URL: https://doi.org/10.1016/S0377-2217(02)00071-1.
  31. Yoichi Iwata, Keigo Oka, and Yuichi Yoshida. Linear-Time FPT Algorithms via Network Flow. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1749-1761. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.127.
  32. Jan Johannsen. Satisfiability Problems Complete for Deterministic Logarithmic Space. In STACS 2004, 21st Annual Symposium on Theoretical Aspects of Computer Science, Montpellier, France, March 25-27, 2004, Proceedings, pages 317-325, 2004. URL: https://doi.org/10.1007/978-3-540-24749-4_28.
  33. Richard M. Karp, Eli Upfal, and Avi Wigderson. Constructing a Perfect Matching is in Random NC. Combinatorica, 6(1):35-48, 1986. URL: https://doi.org/10.1007/BF02579407.
  34. Meena Mahajan and Venkatesh Raman. Parameterizing above Guaranteed Values: MaxSat and MaxCut. J. Algorithms, 31(2):335-354, 1999. URL: https://doi.org/10.1006/jagm.1998.0996.
  35. N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. LP can be a cure for Parameterized Problems. In 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, pages 338-349, 2012. URL: https://doi.org/10.4230/LIPIcs.STACS.2012.338.
  36. George L. Nemhauser and Leslie E. Trotter Jr. Vertex Packings: Structural Properties and Algorithms. Mathematical Programming, 8(1):232-248, 1975. URL: https://doi.org/10.1007/BF01580444.
  37. Arfst Nickelsen and Till Tantau. The Complexity of Finding Paths in Graphs with Bounded Independence Number. SIAM Journal on Computing, 34(5):1176-1195, 2005. URL: https://doi.org/10.1137/S0097539704441642.
  38. Christos H. Papadimitriou and Mihalis Yannakakis. Optimization, Approximation, and Complexity Classes. J. Comput. Syst. Sci., 43(3):425-440, 1991. URL: https://doi.org/10.1016/0022-0000(91)90023-X.
  39. Michal Pilipczuk, Sebastian Siebertz, and Szymon Torunczyk. Parameterized Circuit Complexity of Model-Checking on Sparse Structures. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 789-798. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209136.
  40. Igor Razgon and Barry O'Sullivan. Almost 2-SAT is Fixed-Parameter Tractable. Journal of Computer and System Sciences, 75(8):435-450, 2009. URL: https://doi.org/10.1016/j.jcss.2009.04.002.
  41. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding Odd Cycle Transversals. Operations Research Letters, 32(4):299-301, 2004. URL: https://doi.org/10.1016/j.orl.2003.10.009.
  42. Stefan Szeider. The Parameterized Complexity of k-flip Local Search for SAT and MaxSAT. Discret. Optim., 8(1):139-145, 2011. URL: https://doi.org/10.1016/j.disopt.2010.07.003.
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