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

Abstract

The problem MaxLin2 can be stated as follows. We are given a system S of m equations in variables x 1,?,x n , where each equation $\sum_{i \in I_{j}}x_{i} = b_{j}$ is assigned a positive integral weight w j and $b_{j} \in\mathbb{F}_{2}$ , I j ?{1,2,?,n} for j=1,?,m. We are required to find an assignment of values in $\mathbb{F}_{2}$ to the variables in order to maximize the total weight of the satisfied equations. Let W be the total weight of all equations in S. We consider the following parameterized version of MaxLin2: decide whether there is an assignment satisfying equations of total weight at least W?k, where k is a nonnegative parameter. We prove that this parameterized problem is W[1]-hard even if each equation of S has exactly three variables and every variable appears in exactly three equations and, moreover, each weight w j equals 1 and no two equations have the same left-hand side. We show the tightness of this result by proving that if each equation has at most two variables then the parameterized problem is fixed-parameter tractable. We also prove that if no variable appears in more than two equations then we can maximize the total weight of satisfied equations in polynomial time.