MaxSAT with Absolute Value Functions: A Parameterized Perspective

Authors Max Bannach , Pamela Fleischmann , Malte Skambath



PDF
Thumbnail PDF

File

LIPIcs.SWAT.2022.12.pdf
  • Filesize: 0.8 MB
  • 20 pages

Document Identifiers

Author Details

Max Bannach
  • Institute for Theoretical Computer Science, Universität zu Lübeck, Germany
Pamela Fleischmann
  • Department of Computer Science, Christian-Albrechts-Universität zu Kiel, Germany
Malte Skambath
  • Department of Computer Science, Christian-Albrechts-Universität zu Kiel, Germany

Cite AsGet BibTex

Max Bannach, Pamela Fleischmann, and Malte Skambath. MaxSAT with Absolute Value Functions: A Parameterized Perspective. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SWAT.2022.12

Abstract

The natural generalization of the Boolean satisfiability problem to optimization problems is the task of determining the maximum number of clauses that can simultaneously be satisfied in a propositional formula in conjunctive normal form. In the weighted maximum satisfiability problem each clause has a positive weight and one seeks an assignment of maximum weight. The literature almost solely considers the case of positive weights. While the general case of the problem is only restricted slightly by this constraint, many special cases become trivial in the absence of negative weights. In this work we study the problem with negative weights and observe that the problem becomes computationally harder - which we formalize from a parameterized perspective in the sense that various variations of the problem become W[1]-hard if negative weights are present. Allowing negative weights also introduces new variants of the problem: Instead of maximizing the sum of weights of satisfied clauses, we can maximize the absolute value of that sum. This turns out to be surprisingly expressive even restricted to monotone formulas in disjunctive normal form with at most two literals per clause. In contrast to the versions without the absolute value, however, we prove that these variants are fixed-parameter tractable. As technical contribution we present a kernelization for an auxiliary problem on hypergraphs in which we seek, given an edge-weighted hypergraph, an induced subgraph that maximizes the absolute value of the sum of edge-weights.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • parameterized complexity
  • kernelization
  • weighted maximum satisfiability
  • absolute value maximization

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. N. Alon, G.Z. Gutin, E.J. Kim, S. Szeider, and A. Yeo. Solving max-r-SAT above a tight lower bound. Algorithmica, 61(3):638-655, 2011. Google Scholar
  2. A. Biere, M. Heule, H. van Maaren, and T. Walsh, editors. Handbook of Satisfiability, Second Edition, volume 185 of Front. Artif. Intell. Appl. Press, 2021. Google Scholar
  3. J. Chen, C. Xu, and J. Wang. Dealing with 4-variables by resolution: An improved maxsat algorithm. Theor. Comput. Sci., 670:33-44, 2017. Google Scholar
  4. B. Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. Google Scholar
  5. Y. Crama, O. Ekin, and P.L. Hammer. Variable and term removal from boolean formulae. Discret. Appl. Math., 75(3):217-230, 1997. Google Scholar
  6. R. Crowston, M. Fellows, G. Gutin, M. Jones, F. Rosamond, S. Thomassé, and A. Yeo. Simultaneously Satisfying Linear Equations Over F₂: MaxLin2 and Max-r-Lin2 Parameterized Above Average. In FSTTCS, volume 13, pages 229-240, 2011. Google Scholar
  7. M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  8. H. Dell, E.J. Kim, M. Lampis, V. Mitsou, and T.Mömke. Complexity and approximability of parameterized max-csps. Algorithmica, 79(1):230-250, 2017. Google Scholar
  9. E. Eiben, R. Ganian, D. Knop, and S. Ordyniak. Unary integer linear programming with structural restrictions. In IJCAI, pages 1284-1290. ijcai.org, 2018. Google Scholar
  10. E. Eiben, R. Ganian, D. Knop, and S. Ordyniak. Solving integer quadratic programming via explicit and structural restrictions. In AAAI, pages 1477-1484. AAAI Press, 2019. Google Scholar
  11. F. Eisenbrand, C. Hunkenschröder, K.M. Klein, M. Koutecký, A. Levin, and S. Onn. An algorithmic theory of integer programming, 2019. URL: http://arxiv.org/abs/1904.01361.
  12. P. Erdős and R. Rado. Intersection theorems for systems of sets. J. London Math. Soc., 1(1):85-90, 1960. Google Scholar
  13. J. Flum and M. Grohe. Parameterized Complexity Theory. EATCS. Springer, 2006. Google Scholar
  14. R. Ganian and S. Ordyniak. Solving integer linear programs by exploiting variable-constraint interactions: A survey. Algorithms, 12(12):248, 2019. Google Scholar
  15. R. Ganian and S. Szeider. New width parameters for model counting. In SAT, volume 10491 of Lecture Notes in Computer Science, pages 38-52. Springer, 2017. Google Scholar
  16. M.R. Garey, D.S. Johnson, and L.J. Stockmeyer. Some simplified np-complete graph problems. Theor. Comput. Sci., 1(3):237-267, 1976. Google Scholar
  17. S. Gaspers and S. Szeider. Kernels for global constraints. In IJCAI, pages 540-545. IJCAI/AAAI, 2011. Google Scholar
  18. S. Gaspers and S. Szeider. Guarantees and limits of preprocessing in constraint satisfaction and reasoning. Artif. Intell., 216:1-19, 2014. Google Scholar
  19. T. Gavenčiak, D. Knop, and M. Koutecký. Integer Programming in Parameterized Complexity: Three Miniatures. In IPEC, volume 115, pages 21:1-21:16, 2019. Google Scholar
  20. T. Gavenčiak, M. Koutecký, and D. Knop. Integer programming in parameterized complexity: Five miniatures. Accepted for Discrete Optimization, 2020. URL: https://doi.org/10.1016/j.disopt.2020.100596.
  21. M. Grohe. The structure of tractable constraint satisfaction problems. In MFCS, volume 4162, pages 58-72. Springer, 2006. Google Scholar
  22. K. Jansen, A. Lassota, and L. Rohwedder. Near-linear time algorithm for n-fold ilps via color coding. SIAM J. Discret. Math., 34(4):2282-2299, 2020. Google Scholar
  23. S. Khanna, M. Sudan, and D.P. Williamson. A complete classification of the approximability of maximization problems derived from boolean constraint satisfaction. In STOC, pages 11-20, 1997. Google Scholar
  24. M. Koutecký, A. Levin, and S. Onn. A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs. In ICALP), volume 107 of LIPIcs, pages 85:1-85:14, 2018. Google Scholar
  25. H.W. Lenstra. Integer programming with a fixed number of variables. Math. Oper. Res., 8:538-548, 1983. Google Scholar
  26. M. Mahajan and V. Raman. Parameterizing above guaranteed values: Maxsat and maxcut. J. Algorithms, 31(2):335-354, 1999. Google Scholar
  27. M. Mahajan, V. Raman, and S. Sikdar. Parameterizing above or below guaranteed values. J. Comput. Syst. Sci., 75(2):137-153, 2009. Google Scholar
  28. N. Nishimura, P. Ragde, and S. Szeider. Detecting backdoor sets with respect to horn and binary clauses. In SAT, 2004. Google Scholar
  29. M. Samer and S. Szeider. Algorithms for propositional model counting. J. Discrete Algorithms, 8(1):50-64, 2010. Google Scholar
  30. M. Samer and S. Szeider. Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci., 76(2):103-114, 2010. Google Scholar
  31. T.J. Schaefer. The complexity of satisfiability problems. In STOC, pages 216-226, 1978. Google Scholar
  32. S. Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discret. Optim., 8(1):139-145, 2011. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail