Solving Linear Equations Parameterized by Hamming Weight

Given a system of linear equations A x = b over the binary field F 2 and an integer t ≥ 1 , we study the following three algorithmic problems: Does A x = b have a solution of weight at most t ? Does A x = b have a solution of weight exactly t ? Does A x = b have a solution of weight at least t ? We...

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Vydané v:	Algorithmica Ročník 75; číslo 2; s. 322 - 338
Hlavní autori:	Arvind, V., Köbler, Johannes, Kuhnert, Sebastian, Torán, Jacobo
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2016
Predmet:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation Graph Match Problem Brute Force Search Deterministic Polynomial Time Internal Vertex Logarithmic Space
ISSN:	0178-4617, 1432-0541
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Abstract	Given a system of linear equations A x = b over the binary field F 2 and an integer t ≥ 1 , we study the following three algorithmic problems: Does A x = b have a solution of weight at most t ? Does A x = b have a solution of weight exactly t ? Does A x = 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 A x = b influences the complexity of the problem. We show a sharp dichotomy: for each k ≥ 3 the first two problems are 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.
AbstractList	Given a system of linear equations A x = b over the binary field F 2 and an integer t ≥ 1 , we study the following three algorithmic problems: Does A x = b have a solution of weight at most t ? Does A x = b have a solution of weight exactly t ? Does A x = 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 A x = b influences the complexity of the problem. We show a sharp dichotomy: for each k ≥ 3 the first two problems are 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.
Author	Kuhnert, Sebastian Köbler, Johannes Arvind, V. Torán, Jacobo
Author_xml	– sequence: 1 givenname: V. surname: Arvind fullname: Arvind, V. organization: Institute of Mathematical Sciences – sequence: 2 givenname: Johannes surname: Köbler fullname: Köbler, Johannes organization: Institut für Informatik, Humboldt-Universität zu Berlin – sequence: 3 givenname: Sebastian orcidid: 0000-0003-2197-5803 surname: Kuhnert fullname: Kuhnert, Sebastian email: kuhnert@informatik.hu-berlin.de organization: Institut für Informatik, Humboldt-Universität zu Berlin – sequence: 4 givenname: Jacobo surname: Torán fullname: Torán, Jacobo organization: Institut für Theoretische Informatik, Universität Ulm
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation
Title	Solving Linear Equations Parameterized by Hamming Weight
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