Satisfying more than half of a system of linear equations over GF(2): A multivariate approach

Robert Crowston, Michael Fellows, Gregory Gutin, Mark Jones, E.J Kim, Frances Rosamond, I.Z Ruzsa, S Thomassé, A Yeo

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    In the parameterized problem MaxLin2-AA[k ], we are given a system with variables x1,…,xnx1,…,xn consisting of equations of the form ∏i∈Ixi=b∏i∈Ixi=b, where xi,b∈{−1,1}xi,b∈{−1,1} and I⊆[n]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 least W/2+kW/2+k, where W is the total weight of all equations and k is the parameter (it is always possible for k=0k=0). We show that MaxLin2-AA[k ] has a kernel with at most View the MathML sourceO(k2logk) variables and can be solved in time 2O(klogk)(nm)O(1)2O(klogk)(nm)O(1). This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,rk,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 that Max-r-Lin2-AA[k,rk,r] has a kernel with at most (2k−1)r(2k−1)r variables.
    Original languageEnglish
    Pages (from-to)687-696
    Number of pages10
    JournalJournal of Computer and System Sciences
    Issue number4
    Publication statusPublished - Jun 2014

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