Parameterized Complexity of Satisfying Almost All Linear Equations over $\mathbb{F}_2$

TL;DR

This paper investigates the parameterized complexity of maximizing satisfied equations in systems over GF(2), proving W[1]-hardness in certain cases and fixed-parameter tractability in others.

Contribution

It establishes the W[1]-hardness of MaxLin2 with specific constraints and identifies conditions under which the problem becomes fixed-parameter tractable or polynomial-time solvable.

Findings

W[1]-hardness for systems with exactly three variables per equation and each variable in three equations

Fixed-parameter tractability when each equation has at most two variables

Polynomial-time solvability when no variable appears in more than two equations

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 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.