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Theory of Computing Systems

, Volume 52, Issue 4, pp 719–728 | Cite as

Parameterized Complexity of Satisfying Almost All Linear Equations over \(\mathbb{F}_{2}\)

  • R. Crowston
  • G. Gutin
  • M. Jones
  • A. Yeo
Article
  • 147 Downloads

Abstract

The problem MaxLin2 can be stated as follows. We are given a system S of m equations in variables x 1,…,x n , where each equation \(\sum_{i \in I_{j}}x_{i} = b_{j}\) is assigned a positive integral weight w j and \(b_{j} \in\mathbb{F}_{2}\), I j ⊆{1,2,…,n} for j=1,…,m. We are required to find an assignment of values in \(\mathbb{F}_{2}\) to the variables in order to maximize the total weight of the satisfied equations.

Let W be the total weight of all equations in S. We consider the following parameterized version of MaxLin2: decide whether there is an assignment satisfying equations of total weight at least Wk, where k is a nonnegative parameter. We prove that this parameterized problem is W[1]-hard even if each equation of S has exactly three variables and every variable appears in exactly three equations and, moreover, each weight w j equals 1 and no two equations have the same left-hand side. We show the tightness of this result by proving that if each equation has at most two variables then the parameterized problem is fixed-parameter tractable. We also prove that if no variable appears in more than two equations then we can maximize the total weight of satisfied equations in polynomial time.

Keywords

Parameterized complexity Linear equations Finite field 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Royal HollowayUniversity of LondonEghamUK
  2. 2.University of JohannesburgAuckland ParkSouth Africa

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