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arXiv · 1307.6429

Generalized Wong sequences and their applications to Edmonds' problems

Gábor Ivanyos·Marek Karpinski·Youming Qiao·Miklos Santha
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Abstract

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the n by n matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matrices in B, symbolic determinant identity testing (SDIT) is the question to decide whether there exists a nonsingular matrix in B. The constructive versions of these problems are asking to find a matrix of maximum rank, respectively a nonsingular matrix, if there exists one. Our first algorithm solves the constructive SMR when B is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when B is spanned by triangularizable matrices, but the triangularization is not given explicitly. Both algorithms work over finite fields of size at least n+1 and over the rational numbers, and the first algorithm actually solves (the non-constructive) SMR independently from the field size. Our main tool to obtain these results is to generalize Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces.

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BibTeXRIS

Gábor Ivanyos, Marek Karpinski, Youming Qiao, Miklos Santha. 2014-06-26. Generalized Wong sequences and their applications to Edmonds' problems. https://arxiv.org/abs/1307.6429

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