Solving Linear Equations Parameterized by Hamming Weight - Arvind, V.; Köbler, Johannes; Kuhnert, Sebastian; Torán, Jacobo (Algorithmica, 2016)
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.