Parameterized Complexity of Satisfying Almost All Linear Equations over   $\mathbb{F}_2$
  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 is $\sum_{i \in I_j}x_i = b_j$ is assigned a positive integral weight $w_j$ and $x_i,b_j \in \mathbb{F}_2$, $I_j \subseteq \{1,2,...,n\}$ for $j=1,...,m$. We are required to find an assignment of values 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. 
R. Crowston, G. Gutin, M. Jones, and A. Yeo
cs.CC cs.DM cs.DS