Solving MAX-r-SAT above a tight lower bound

  • Authors:

  • Noga Alon;Gregory Gutin;Eun Jung Kim;Stefan Szeider;Anders Yeo

  • Affiliations:

  • Tel Aviv University, Tel Aviv, Israel;University of London, Egham, Surrey, UK;University of London, Egham, Surrey, UK;Vienna University of Technology, Vienna, Austria;University of London, Egham, Surrey, UK

  • Venue:
  • SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
  • Year:
  • 2010

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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)/2r 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.