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

Authors：Crowston R; Fellows M; Gutin G; Jones M; Kim E J; Rosamond F; Ruzsa I Z; Thomasse S; Yeo A

Source：Journal of Computer and System Sciences, 2014, 80(4): 687-696.

DOI：10.1016/j.jcss.2013.10.002

• Fixed-parameter tractable
• Kernel
• MaxLin

In the parameterized problem MaxLin2-AA[k], we are given a system with variables x1,.,xn consisting of equations of the form ∝iεI xi=b, where xi,bε{-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.

• Institution
  Singapore University of Technology and Design