TY - JOUR

T1 - Matrix rigidity from the viewpoint of parameterized complexity∗

AU - Fomin, Fedor V.

AU - Lokshtanov, Daniel

AU - Meesum, S. M.

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - Funding Information:
∗Received by the editors March 27, 2017; accepted for publication (in revised form) December 22, 2017; published electronically May 1, 2018. A preliminary version of this article has appeared in the Proceedings of STACS 2017. http://www.siam.org/journals/sidma/32-2/M112258.html Funding: This work was funded by European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant 306992. †Department of Informatics, University of Bergen, Bergen, Norway (fomin@ii.uib.no, daniello@ii. uib.no, meirav.zehavi@ii.uib.no). ‡Theoretical Computer Science, The Institute of Mathematical Sciences, HBNI, Chennai, Tamil Nadu 600113, India (meesum@imsc.res.in). §Department of Informatics, University of Bergen, Bergen, Norway and Theoretical Computer Science, The Institute of Mathematical Sciences, HBNI, Chennai, Tamil Nadu 600113, India (saket@ imsc.res.in).
Funding Information:
This work was funded by European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant 306992.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Linear algebra

KW - Matrix rigidity

KW - Parameterized complexity

UR - http://www.scopus.com/inward/record.url?scp=85049562314&partnerID=8YFLogxK

U2 - 10.1137/17M112258X

DO - 10.1137/17M112258X

M3 - Article

AN - SCOPUS:85049562314

SN - 0895-4801

VL - 32

SP - 966

EP - 985

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 2

ER -