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 nxn 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, while 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 independent of 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.
@InProceedings{ivanyos_et_al:LIPIcs.STACS.2014.397, author = {Ivanyos, G\'{a}bor and Karpinski, Marek and Qiao, Youming and Santha, Miklos}, title = {{Generalized Wong sequences and their applications to Edmonds' problems}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {397--408}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.397}, URN = {urn:nbn:de:0030-drops-44741}, doi = {10.4230/LIPIcs.STACS.2014.397}, annote = {Keywords: symbolic determinantal identity testing, Edmonds' problem, maximum rank matrix completion, derandomization, Wong sequences} }
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