Some optimal inapproximability results
Journal of the ACM (JACM)
Discrete Applied Mathematics
Algorithms with large domination ratio
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)
Some topics in analysis of boolean functions
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Parameterizing above or below guaranteed values
Journal of Computer and System Sciences
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
Extremal Combinatorics: With Applications in Computer Science
Extremal Combinatorics: With Applications in Computer Science
Parameterizing MAX SNP problems above guaranteed values
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Parameterized Complexity
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
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
New lower bound on max cut of hypergraphs with an application to r-set splitting
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Kernelization --- preprocessing with a guarantee
The Multivariate Algorithmic Revolution and Beyond
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 bound for 3-satisfiable MaxSat and its algorithmic application
Information and Computation
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
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In the problem Max Lin, we are given a system Az=b of m linear equations with n variables over $\mathbb{F}_2$ in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least k, where k is the parameter. It is not hard to see that we may assume that no two equations in Az=b have the same left-hand side and n=rank A. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: m≤2p(n) for an arbitrary fixed function p(n)=o(n). This result generalizes earlier results by Crowston et al. (arXiv:0911.5384) and Gutin et al. (Proc. IWPEC'09). We also prove that Max Lin AA is polynomial-time solvable for every fixed k and, moreover, Max Lin AA is in the parameterized complexity class W[P]. Max r-Lin AA is a special case of Max Lin AA, where each equation has at most r variables. In Max Exact r-SAT AA we are given a multiset of m clauses on n variables such that each clause has r variables and asked whether there is a truth assignment to the n variables that satisfies at least (1−2−r)m+k2−r clauses. Using our maximum excess results, we prove that for each fixed r≥2, Max r-Lin AA and Max Exact r-SAT AA can be solved in time 2O(k logk)+mO(1). This improves $2^{O(k^2)}+m^{O(1)}$-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively. It is easy to see that maximization of arbitrary pseudo-boolean functions, i.e., functions $f:\ \{-1,+1\}^n\rightarrow \mathbb{R}$, represented by their Fourier expansions is equivalent to solving Max Lin. Using our main maximum excess result, we obtain a tight lower bound on the maxima of pseudo-boolean functions.