The Computational Complexity of Some Problems of Linear Algebra

Abstract

We consider the computational complexity of some problems dealing with matrix rank.<br /> Let E, S be subsets of a commutative ring R.<br />Let x1, x2, ..., xt be variables. Given a matrix M = M(x1, x2, ..., xt)<br />with entries chosen from E union {x1, x2, ..., xt}, we want to determine<br />maxrankS(M) = max rank M(a1, a2, ... , at)<br />and<br />minrankS(M) = min rank M(a1, a2, ..., at). <br />There are also variants of these problems that specify more about the<br />structure of M, or instead of asking for the minimum or maximum rank, <br />ask if there is some substitution of the variables that makes the matrix<br /> invertible or noninvertible.<br />Depending on E, S, and on which variant is studied, the complexity<br />of these problems can range from polynomial-time solvable to random<br />polynomial-time solvable to NP-complete to PSPACE-solvable to<br />unsolvable.

Two Attacks on Rank Metric Code-Based Schemes: RankSign and an IBE Scheme

“…Finding rank 1 matrices in C proj pub by solving a bilinear system The basic bilinear system. Finding rank 1 matrices in C proj pub can be formulated as an instance of the MinRank problem [BFS99,Cou01]. We could use standard techniques for solving this problem [KS99, FLdVP08, FDS10, Spa12] but we found that it is better here to use the algebraic modelling suggested in [AGH + 17].…”

“…The basic bilinear system. Finding codewords of rank w dec in C Mat can be expressed as an instance of the MinRank problem [BFS99,Cou01]. Once again we propose the algebraic modeling which was suggested in [AGH + 17].…”

“…This depends of course on how N is viewed as a product of two elements. Decoding in this metric is known to be an NP hard problem [BFS99,Cou01]. In the particular case of [GRSZ14], the codes which are considered are not F q -linear but, as is customary in the setting of rank metric based cryptography, F q m -linear: the codes are here subspaces of F n q m .…”

Zero forcing in iterated line digraphs

“…In addition to its intrinsic relation to minimum rank, zero forcing is closely related to power domination, a graph theory concept introduced in [20] to optimize the monitoring process of electrical power networks. From the definitions of power domination and zero forcing, it follows that the closed out neighborhood of a power dominating set is a zero forcing set, and a stronger relationship between zero forcing and power domination was established in [7].Zero forcing, minimum rank and power domination are all NP-hard problems, as proven in [1], [10], and [20], respectively. Thus, it is important to obtain bounds for the minimum rank, the zero forcing and the power domination numbers, as well as closed formulas to calculate them for families of graphs.…”

“…Zero forcing, minimum rank and power domination are all NP-hard problems, as proven in [1], [10], and [20], respectively. Thus, it is important to obtain bounds for the minimum rank, the zero forcing and the power domination numbers, as well as closed formulas to calculate them for families of graphs.…”