Matrix rigidity from the viewpoint of parameterized complexity

Fedor V. Fomin, Daniel Lokshtanov, S. M. Meesum, Saket Saurabh, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

The rigidity of a matrix A for a target rank r over a field F is the minimum Hamming distance between A and a matrix of rank at most r. Rigidity is a classical concept in Computational Complexity Theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits. Yet, from the viewpoint of Parameterized Complexity, the study of central properties of matrices in general, and of the rigidity of a matrix in particular, has been neglected. In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of Parameterized Complexity. Naturally, given parameters r and k, the MATRIX RIGIDITY problem asks whether the rigidity of A for the target rank r is at most k. We show that in case D = ℝ or F is any finite field, this problem is fixed-parameter tractable with respect to k + r. To this end, we present a dimension reduction procedure, which may be a valuable primitive in future studies of problems of this nature. We also employ central tools in Real Algebraic Geometry, which are not well known in Parameterized Complexity, as a black box. In particular, we view the output of our dimension reduction procedure as an algebraic variety. Our main results are complemented by a W[1]-hardness result and a subexponential-time parameterized algorithm for a special case of MATRIX RIGIDITY, highlighting the different flavors of this problem.

Original languageEnglish
Title of host publication34th Symposium on Theoretical Aspects of Computer Science, STACS 2017
EditorsBrigitte Vallee, Heribert Vollmer
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770286
DOIs
StatePublished - 1 Mar 2017
Externally publishedYes
Event34th Symposium on Theoretical Aspects of Computer Science, STACS 2017 - Hannover, Germany
Duration: 8 Mar 201711 Mar 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume66
ISSN (Print)1868-8969

Conference

Conference34th Symposium on Theoretical Aspects of Computer Science, STACS 2017
Country/TerritoryGermany
CityHannover
Period8/03/1711/03/17

Keywords

  • Linear algebra
  • Matrix rigidity
  • Parameterized complexity

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