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Solving Linear Equations Parameterized by Hamming Weight

  • Vikraman Arvind
  • Johannes KöblerEmail author
  • Sebastian Kuhnert
  • Jacobo Torán
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)

Abstract

Given a system of linear equations \(Ax=b\) over the binary field \(\mathbb {F}_2\) and an integer \(t\ge 1\), we study the following three algorithmic problems:
  1. 1.

    Does \(Ax=b\) have a solution of weight at most \(t\)?

     
  2. 2.

    Does \(Ax=b\) have a solution of weight exactly \(t\)?

     
  3. 3.

    Does \(Ax=b\) have a solution of weight at least \(t\)?

     
We investigate the parameterized complexity of these problems with \(t\) as parameter. A special aspect of our study is to show how the maximum multiplicity \(k\) of variable occurrences in \(Ax=b\) influences the complexity of the problem. We show a sharp dichotomy: for each \(k\ge 3\) the first two problems are Open image in new window -hard (which strengthens and simplifies a result of Downey et al. [SIAM J. Comput. 29, 1999]). For \(k=2\), the problems turn out to be intimately connected to well-studied matching problems and can be efficiently solved using matching algorithms.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Vikraman Arvind
    • 1
  • Johannes Köbler
    • 2
    Email author
  • Sebastian Kuhnert
    • 2
  • Jacobo Torán
    • 3
  1. 1.Institute of Mathematical SciencesChennaiIndia
  2. 2.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany
  3. 3.Institut für Theoretische InformatikUniversität UlmUlmGermany

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