Matrix rigidity from the viewpoint of parameterized complexity

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

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

For a target rank r, the rigidity of a matrix A 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 the case F = 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
Pages (from-to)966-985
Number of pages20
JournalSIAM Journal on Discrete Mathematics
Volume32
Issue number2
DOIs
StatePublished - 1 Jan 2018
Externally publishedYes

Keywords

  • Linear algebra
  • Matrix rigidity
  • Parameterized complexity

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