文章基本信息

标题：Matrix Rigidity from the Viewpoint of Parameterized Complexity
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作者：Fedor V. Fomin ; Daniel Lokshtanov ; S. M. Meesum 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：66
页码：32:1-32:14
DOI：10.4230/LIPIcs.STACS.2017.32
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：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 F equals the reals 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.
关键词：Matrix Rigidity; Parameterized Complexity; Linear Algebra