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Simultaneously Satisfying Linear Equations Over $\mathbb{F}_2$: MaxLin2 and Max-$r$-Lin2 Parameterized Above Average

In the parameterized problem \textsc{MaxLin2-AA}[$k$], we are given a system with variables $x_1,...,x_n$ consisting of equations of the form $\prod_{i \in I}x_i = b$, where $x_i,b \in \{-1, 1\}$ and $I\subseteq [n],$ each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least $W/2+k$, where $W$ is the total weight of all equations and $k$ is the parameter (if $k=0$, the possibility is assured). We show that \textsc{MaxLin2-AA}[$k$] has a kernel with at most $O(k^2\log k)$ variables and can be solved in time $2^{O(k\log k)}(nm)^{O(1)}$. This solves an open problem of Mahajan et al. (2006). The problem \textsc{Max-$r$-Lin2-AA}[$k,r$] is the same as \textsc{MaxLin2-AA}[$k$] with two differences: each equation has at most $r$ variables and $r$ is the second parameter. We prove a theorem on \textsc{Max-$r$-Lin2-AA}[$k,r$] which implies that \textsc{Max-$r$-Lin2-AA}[$k,r$] has a kernel with at most $(2k-1)r$ variables improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function $f:\ \{-1,1\}^n \rightarrow \mathbb{R}$ of degree $r$. We show applicability of the lower bound by giving a new proof of the Edwards-Erd{\H o}s bound (each connected graph on $n$ vertices and $m$ edges has a bipartite subgraph with at least $m/2 + (n-1)/4$ edges) and obtaining a generalization.