Simultaneously Satisfying Linear Equations Over F_2: MaxLin2 and Max-r-Lin2 Parameterized Above Average

In the parameterized problem MAXLIN2-AA[k], we are given a system with variables x1, . . . , xn consisting of equations of the form ∏ i∈I xi = b, where xi, b ∈ {−1, 1} and I ⊆ [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 leastW/2+ k, whereW is the total weight of all equations and k is the parameter (if k = 0, the possibility is assured). We show that MAXLIN2AA[k] has a kernel with at most O(k log k) variables and can be solved in time 2 log (nm). This solves an open problem of Mahajan et al. (2006). The problem MAX-r-LIN2-AA[k, r] is the same as MAXLIN2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on MAX-r-LIN2-AA[k, r] which implies that MAX-rLIN2-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} → R of degree r. We show applicability of the lower bound by giving a new proof of the EdwardsErdős 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.