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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Bibliographic Details
Published in:Algorithmica Vol. 75; no. 2; pp. 322 - 338
Main Authors: Arvind, V., Köbler, Johannes, Kuhnert, Sebastian, Torán, Jacobo
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2016
Subjects:
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
Online Access:Get full text
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Summary: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.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-015-0098-3