Parameterizing above guaranteed values: MaxSat and MaxCut
Journal of Algorithms
On the advantage over a random assignment
Random Structures & Algorithms
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
The Linear Arrangement Problem Parameterized Above Guaranteed Value
Theory of Computing Systems
Fixed-Parameter Complexity of Minimum Profile Problems
Algorithmica - Parameterized and Exact Algorithms
Parameterizing above or below guaranteed values
Journal of Computer and System Sciences
Interval Completion Is Fixed Parameter Tractable
SIAM Journal on Computing
A Probabilistic Approach to Problems Parameterized above or below Tight Bounds
Parameterized and Exact Computation
Solving MAX-r-SAT above a tight lower bound
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Note on maximal bisection above tight lower bound
Information Processing Letters
Betweenness parameterized above tight lower bound
Journal of Computer and System Sciences
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
A probabilistic approach to problems parameterized above or below tight bounds
Journal of Computer and System Sciences
A new bound for 3-satisfiable maxsat and its algorithmic application
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Journal of Computer and System Sciences
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
Satisfying more than half of a system of linear equations over GF(2): A multivariate approach
Journal of Computer and System Sciences
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In the Max Lin-2 problem we are given a system S of m linear equations in n variables over F"2 in which equation j is assigned a positive integral weight w"j for each j. We wish to find an assignment of values to the variables which maximizes the total weight of satisfied equations. This problem generalizes Max Cut. The expected weight of satisfied equations is W/2, where W=w"1+...+w"m; W/2 is a tight lower bound on the optimal solution of Max Lin-2. Mahajan et al. (Parameterizing above or below guaranteed values, J. Comput. Syst. Sci. 75 (2009) 137-153) stated the following parameterized version of Max Lin-2: decide whether there is an assignment of values to the variables that satisfies equations of total weight at least W/2+k, where k is the parameter. They asked whether this parameterized problem is fixed-parameter tractable, i.e., can be solved in time f(k)(nm)^O^(^1^), where f(k) is an arbitrary computable function in k only. Their question remains open, but using some probabilistic inequalities and, in one case, a Fourier analysis inequality, Gutin et al. (A probabilistic approach to problems parameterized above tight lower bound, in: Proc. IWPEC'09, in: Lect. Notes Comput. Sci., vol. 5917, 2009, pp. 234-245) proved that the problem is fixed-parameter tractable in three special cases. In this paper we significantly extend two of the three special cases using only tools from combinatorics. We show that one of our results can be used to obtain a combinatorial proof that another problem from Mahajan et al. (Parameterizing above or below guaranteed values, J. Comput. Syst. Sci. 75 (2009) 137-153), Max r-SAT above Average, is fixed-parameter tractable for each r=2. Note that Max r-SAT above Average has been already shown to be fixed-parameter tractable by Alon et al. (Solving MAX-r-SAT above a tight lower bound, in: Proc. SODA 2010, pp. 511-517), but the paper used the approach of Gutin et al. (A probabilistic approach to problems parameterized above tight lower bound, in: Proc. IWPEC'09, in: Lect. Notes Comput. Sci., vol. 5917, 2009, pp. 234-245).