{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7521","name":"Generalized Wong sequences and their applications to Edmonds' problems","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 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.\r\n\r\nOur 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.","keywords":["symbolic determinantal identity testing","Edmonds' problem","maximum rank matrix completion","derandomization","Wong sequences"],"author":[{"@type":"Person","name":"Ivanyos, G\u00e1bor","givenName":"G\u00e1bor","familyName":"Ivanyos"},{"@type":"Person","name":"Karpinski, Marek","givenName":"Marek","familyName":"Karpinski"},{"@type":"Person","name":"Qiao, Youming","givenName":"Youming","familyName":"Qiao"},{"@type":"Person","name":"Santha, Miklos","givenName":"Miklos","familyName":"Santha"}],"position":35,"pageStart":397,"pageEnd":408,"dateCreated":"2014-03-05","datePublished":"2014-03-05","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ivanyos, G\u00e1bor","givenName":"G\u00e1bor","familyName":"Ivanyos"},{"@type":"Person","name":"Karpinski, Marek","givenName":"Marek","familyName":"Karpinski"},{"@type":"Person","name":"Qiao, Youming","givenName":"Youming","familyName":"Qiao"},{"@type":"Person","name":"Santha, Miklos","givenName":"Miklos","familyName":"Santha"}],"copyrightYear":"2014","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2014.397","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6228","volumeNumber":25,"name":"31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)","dateCreated":"2014-03-05","datePublished":"2014-03-05","editor":[{"@type":"Person","name":"Mayr, Ernst W.","givenName":"Ernst W.","familyName":"Mayr"},{"@type":"Person","name":"Portier, Natacha","givenName":"Natacha","familyName":"Portier"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7521","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6228"}}}