Abstract
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 ((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 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(9r 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(9r 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.
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A preliminary version of this paper has appeared in the proceedings of ACM-SIAM Symposium on Discrete Algorithms (SODA 2010). We extend the preliminary version by introducing the notion of a bikernelization and using it to prove the existence of a polynomial kernel for the Max-r-CSP tlb problem introduced in Sect. 6. We also obtain a quadratic kernel for Max-r-Sat tlb .
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Alon, N., Gutin, G., Kim, E.J. et al. Solving MAX-r-SAT Above a Tight Lower Bound. Algorithmica 61, 638–655 (2011). https://doi.org/10.1007/s00453-010-9428-7
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DOI: https://doi.org/10.1007/s00453-010-9428-7