Improved parameterized algorithms for above average constraint satisfaction
IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
Parameterized complexity of maxsat above average
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Discrete Optimization
Constraint satisfaction problems parameterized above or below tight bounds: a survey
The Multivariate Algorithmic Revolution and Beyond
Communication: Hypercontractive inequality for pseudo-Boolean functions of bounded Fourier width
Discrete Applied Mathematics
Parameterized study of the test cover problem
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
A new algorithm for parameterized MAX-SAT
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
Homomorphic hashing for sparse coefficient extraction
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
Bisections above tight lower bounds
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
A new bound for 3-satisfiable MaxSat and its algorithmic application
Information and Computation
Preprocessing subgraph and minor problems: When does a small vertex cover help?
Journal of Computer and System Sciences
Parameterized complexity of MaxSat Above Average
Theoretical Computer Science
Satisfying more than half of a system of linear equations over GF(2): A multivariate approach
Journal of Computer and System Sciences
Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width
Fundamenta Informaticae - MFCS & CSL 2010 Satellite Workshops: Selected Papers
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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.