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

Crowston, Robert; Fellows, Michael R.; Gutin, Gregory; Jones, Mark; Kim, Eun Jung; Rosamond, Frances; Rusza, Imre Z.; Thomassé, Stéphan; Yeo, Anders (2014), Satisfying more than half of a system of linear equations over GF(2): A multivariate approach, Journal of Computer and System Sciences, 80, 4, p. 687-696. 10.1016/j.jcss.2013.10.002

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Type
Article accepté pour publication ou publié
Date
2014
Journal name
Journal of Computer and System Sciences
Volume
80
Number
4
Publisher
Elsevier
Pages
687-696
Publication identifier
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Author(s)
Crowston, Robert
Department of Computer Science [Royal Holloway]
Fellows, Michael R.
Autre
Gutin, Gregory
Department of Computer Science
Jones, Mark
Department of Computer Science
Kim, Eun Jung
Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Rosamond, Frances
Autre
Rusza, Imre Z.
Alfréd Rényi Institute of Mathematics
Thomassé, Stéphan
Laboratoire de l'Informatique du Parallélisme [LIP]
Yeo, Anders
Singapore University of Technology and Design [SUTD]
Abstract (EN)
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.
Subjects / Keywords
MaxLin; Kernel; Fixed-parameter tractable