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.

💡 Research Summary

The paper investigates a parameterized version of the classic MaxLin2 problem, which asks for an assignment to Boolean variables that maximizes the total weight of satisfied linear equations over the field ℱ₂. Given a system S of m equations ∑_{i∈I_j} x_i = b_j with positive integer weights w_j, let W = Σ_j w_j be the total weight. The parameterized problem MaxLin2‑B