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

• Published in: Journal of computer and system sciences Vol. 80; no. 4; pp. 687 - 696
• Main Authors: Crowston, R., Fellows, M., Gutin, G., Jones, M., Kim, E.J., Rosamond, F., Ruzsa, I.Z., Thomassé, S., Yeo, A.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.06.2014; Elsevier
• Subjects: Computer Science; Data Structures and Algorithms; Fixed-parameter tractable; Integrals; Kernel; Kernels; Linear equations; Mathematical analysis; Mathematical models; MaxLin; Kernel; MaxLin; Fixed-parameter tractable
• ISSN: 0022-0000; 1090-2724
• DOI: 10.1016/j.jcss.2013.10.002

In the parameterized problem MaxLin2-AA[k], we are given a system with variables x1,…,xn consisting of equations of the form ∏i∈Ixi=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 least W/2+k, where W is the total weight of all equations and k is the parameter (it is always possible for k=0). We show that MaxLin2-AA[k] has a kernel with at most O(k2logk) variables and can be solved in time 2O(klogk)(nm)O(1). 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 that Max-r-Lin2-AA[k,r] has a kernel with at most (2k−1)r variables. •Can we satisfy more than half of a system of linear equations over GF(2)?•Parameter is the difference between the number of satisfied and falsified equations.•We provide a kernel and a fixed-parameter algorithm for the parameterized problem.•This answers an open question of Mahajan et al. (2006).