Systems of Linear Equations over $\mathbb{F}_2$ and Problems Parameterized above Average

Abstract

In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over F2 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 = rankA. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: m ≤ 2 p(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 2 It is easy to see that maximization of arbitrary pseudo-boolean functions, i.e., functions f : {−1, +1} n → 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.

Parameterized Study of the Test Cover Problem

“…In this paper we carry out a systematic study of Test Cover in the realm of parameterized complexity 4 . The following will be a generic parameterization of the problem: p(k, n, m)-Test Cover Instance: A set T of m tests on [n] such that T is a test cover.…”

“…In the same paper a quadratic kernel was also given for Max r-SAT. Other significant results in this direction include quadratic kernels for ternary permutation constraint satisfaction problems parameterized above average and results on systems of linear equations over the field of two elements [3,4,10].…”

A probabilistic approach to problems parameterized above or below tight bounds

“…Using different approaches, Crowston et al [7,8] obtained significant extensions of Cases 2 and 3 of Theorem 2.…”

“…The remark before the lemma immediately implies that T can be obtained from S in time polynomial in n and m. The rest of the proof is taken from [8]. Consider an independent set I of columns of A of cardinality rank A and a column a j / ∈ I .…”

Parameterized Complexity of Satisfying Almost All Linear Equations over $\mathbb{F}_{2}$

“…Using a probabilistic approach, Gutin et al [9] proved that Max-r-Lin2-AA admits a kernel with a quadratic number O(k 2 ) of variables and equations and, thus, is fixed-parameter tractable. With respect to the number of variables, this result was improved by Crowston et al [4] to O(k log k) and by Kim and Williams [11] to O(k). The parameterized complexity of MaxLin2-AA was established in a series of two papers [4,3], where it was proved, using a combination of algorithmic and linear-algebraic techniques, that MaxLin2-AA admits a kernel with at most O(k 2 log k) variables 2 and, thus, MaxLin2-AA is fixed-parameter tractable.…”