Systems of Linear Equations over $\mathbb{F}_2$ and Problems Parameterized Above Average
R. Crowston, G. Gutin, M. Jones, E.J. Kim and I. Ruzsa
Algorithmica 2010.

## Abstract

In the problem Max Lin, we are given a system $Az=b$ of $m$ linear equations with $n$ variables over $\mathbb{F}_2$ 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 $n={\rm rank A}$. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: $m\le 2^{p(n)}$ 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 $r\ge 2$, Max $r$-Lin AA and Max Exact $r$-SAT AA can be solved in time $2^{O(k \log k)}+m^{O(1)}.$ This improves $2^{O(k^2)}+m^{O(1)}$-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively. It is easy to see that maximization of arbitrary pseudo-boolean functions, i.e., functions $f:\ \{-1,+1\}^n\rightarrow \mathbb{R}$, 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.