Solving MAX-r-SAT above a tight lower bound

Noga Alon, Gregory Gutin, Eun Jung Kim, Stefan Szeider, Anders Yeo

Research output: Chapter in Book/Report/Conference proceedingConference contribution

22 Scopus citations

Abstract

We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2r - 1)m + k)/2 r clauses. Thus MAX-r-SAT is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1 - 2-r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k2), then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of MAX-2-SAT with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k)/4 clauses. We also outline how the fixed-parameter tractability result on MAX-r-SAT can be extended to a family of Boolean Constraint Satisfaction Problems.

Original languageEnglish (US)
Title of host publicationProceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherAssociation for Computing Machinery (ACM)
Pages511-517
Number of pages7
ISBN (Print)9780898717013
DOIs
StatePublished - Jan 1 2010
Externally publishedYes
Event21st Annual ACM-SIAM Symposium on Discrete Algorithms - Austin, TX, United States
Duration: Jan 17 2010Jan 19 2010

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other21st Annual ACM-SIAM Symposium on Discrete Algorithms
CountryUnited States
CityAustin, TX
Period1/17/101/19/10

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Alon, N., Gutin, G., Kim, E. J., Szeider, S., & Yeo, A. (2010). Solving MAX-r-SAT above a tight lower bound. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 511-517). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery (ACM). https://doi.org/10.1137/1.9781611973075.44