, Volume 61, Issue 3, pp 638–655 | Cite as

Solving MAX-r-SAT Above a Tight Lower Bound

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


We present an exact algorithm that decides, for every fixed r≥2 in time \(O(m)+2^{O(k^{2})}\) whether a given multiset of m clauses of size r admits a truth assignment that satisfies at least ((2 r −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 (1−2r )m. This solves an open problem of Mahajan et al. (J. Comput. Syst. Sci. 75(2):137–153, 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(9 r k 2) 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(9 r k 2), then there is a truth assignment satisfying the required number of clauses.

We introduce a new notion of bikernelization from a parameterized problem to another one and apply it to prove that the above-mentioned parameterized Max-r-Sat admits a polynomial-size kernel.

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 a certain polynomial time reduction rule to it, then there is a truth assignment that satisfies at least (3m+k)/4 clauses.

We also outline how the fixed-parameter tractability and polynomial-size kernel results on Max-r-Sat can be extended to more general families of Boolean Constraint Satisfaction Problems.


Max SAT Fixed-parameter tractable Above lower bound Kernel Bikernel 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Noga Alon
    • 1
  • Gregory Gutin
    • 2
  • Eun Jung Kim
    • 2
  • Stefan Szeider
    • 3
  • Anders Yeo
    • 2
  1. 1.Schools of Mathematics and Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Computer Science, Royal HollowayUniversity of LondonEghamUK
  3. 3.Institute of Information SystemsVienna University of TechnologyViennaAustria

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