The complexity of matrix rank and feasible systems of linear equations

Eric Allender, Robert Beals, Mitsunori Ogihara

Research output: Chapter in Book/Report/Conference proceedingConference contribution

10 Scopus citations

Abstract

We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix, (as well as other problems), are complete under logspace reductions. As an important part of presenting this classification, we show that the "exact counting logspace hierarchy" collapses to near the bottom level. (We review the definition of this hierarchy below.) We further show that this class is closed under NC1-reducibihty, and that it consists of exactly those languages that have logspace uniform span programs (introduced by Karchmer and Wigderson) over the rationals. In addition, we contrast the complexity of these problems with the complexity of determining if a system of linear equations has an integer solution.

Original languageEnglish (US)
Title of host publicationProceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996
PublisherAssociation for Computing Machinery
Pages161-167
Number of pages7
ISBN (Electronic)0897917855
DOIs
StatePublished - Jul 1 1996
Event28th Annual ACM Symposium on Theory of Computing, STOC 1996 - Philadelphia, United States
Duration: May 22 1996May 24 1996

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
VolumePart F129452
ISSN (Print)0737-8017

Conference

Conference28th Annual ACM Symposium on Theory of Computing, STOC 1996
CountryUnited States
CityPhiladelphia
Period5/22/965/24/96

ASJC Scopus subject areas

  • Software

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  • Cite this

    Allender, E., Beals, R., & Ogihara, M. (1996). The complexity of matrix rank and feasible systems of linear equations. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, STOC 1996 (pp. 161-167). (Proceedings of the Annual ACM Symposium on Theory of Computing; Vol. Part F129452). Association for Computing Machinery. https://doi.org/10.1145/237814.237856