Implications of forbidden structures for extremal algorithmic problems
Theoretical Computer Science
Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas
Journal of Combinatorial Theory Series A
On the approximation of maximum satisfiability
SODA selected papers from the third annual ACM-SIAM symposium on Discrete algorithms
Parameterizing above guaranteed values: MaxSat and MaxCut
Journal of Algorithms
Complexity of Partial Satisfaction
Journal of the ACM (JACM)
Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference
Theoretical Computer Science
Lean clause-sets: generalizations of minimally unsatisfiable clause-sets
Discrete Applied Mathematics - The renesse issue on satisfiability
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
On Local Versus Global Satisfiability
SIAM Journal on Discrete Mathematics
Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
Journal of Computer and System Sciences
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterizing above or below guaranteed values
Journal of Computer and System Sciences
Lower Bounds for Kernelizations and Other Preprocessing Procedures
CiE '09 Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice
Solving satisfiability in less than 2n steps
Discrete Applied Mathematics
Note on Max Lin-2 above Average
Information Processing Letters
Solving MAX-r-SAT above a tight lower bound
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Systems of linear equations over F2 and problems parameterized above average
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Kernels: annotated, proper and induced
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Improved parameterized algorithms for above average constraint satisfaction
IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
Parameterized Complexity
Parameterized complexity of maxsat above average
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Constraint satisfaction problems parameterized above or below tight bounds: a survey
The Multivariate Algorithmic Revolution and Beyond
A new bound for 3-satisfiable MaxSat and its algorithmic application
Information and Computation
Parameterized complexity of MaxSat Above Average
Theoretical Computer Science
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Let F be a CNF formula with n variables and m clauses. F is tsatisfiable if for any t clauses in F, there is a truth assignment which satisfies all of them. Lieberherr and Specker (1982) and, later, Yannakakis (1994) proved that in each 3-satisfiable CNF formula at least 2/3 of its clauses can be satisfied by a truth assignment. Yannakakis's proof utilizes the fact that 2/3m is a lower bound on the expected number of clauses satisfied by a random truth assignment over a certain distribution. A CNF formula F is called expanding if for every subset X of the variables of F, the number of clauses containing variables of X is not smaller than |X|. In this paper we strengthen the 2/3m bound for expanding 3- satisfiable CNF formulas by showing that for every such formula F at least 2 3m+ρn clauses of F can be satisfied by a truth assignment, where ρ( 0.0019) is a constant. Our proof uses a probabilistic method with a sophisticated distribution for truth values. We use the bound 2 3m + ρn and results on matching autarkies to obtain a new lower bound on the maximum number of clauses that can be satisfied by a truth assignment in any 3-satisfiable CNF formula. We use our results above to show that the following parameterized problem is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. In 3-S-MAXSAT-AE, we are given a 3-satisfiable CNF formula F with m clauses and asked to determine whether there is an assignment which satisfies at least 2 3m + k clauses, where k is the parameter. Note that Mahajan and Raman (1999) asked whether 2-S-MAXSAT-AE, the corresponding problem for 2-satisfiable formulas, is fixed-parameter tractable. Crowston and the authors of this paper proved in [9] that 2-S-MAXSAT-AE is fixed-parameter tractable and, moreover, has a kernel with a linear number of variables. 2-S-MAXSAT-AE appears to be easier than 3-S-MAXSAT-AE and, unlike this paper, [9] uses only deterministic combinatorial arguments.