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Algorithmica

, Volume 75, Issue 2, pp 322–338 | Cite as

Solving Linear Equations Parameterized by Hamming Weight

  • V. Arvind
  • Johannes Köbler
  • Sebastian Kuhnert
  • Jacobo Torán
Article

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 \(\textsf {W[1] }\)-hard [which strengthens and simplifies a result of Downey et al. (SIAM J Comput 29(2), 545–570, 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.

Keywords

Internal Vertex Logarithmic Space Brute Force Search Deterministic Polynomial Time Graph Match Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

Gaurav Rattan and Oleg Verbitsky suggested and proved Lemma 6.1, which allowed us to improve the fpt algorithm for \(\textsc {LinEq}_{\ge {,t}}\) of the conference version to the polynomial time kernelization of Lemma 6.2. We also thank the reviewers for their helpful remarks.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  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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