Complexity of matrix rank and feasible systems of linear equations (extended abstract)

Eric Allender, Robert Beals, Mitsunori Ogihara

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

10 Citations (Scopus)

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-reducibility, 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 publicationConference Proceedings of the Annual ACM Symposium on Theory of Computing
PublisherACM
Pages161-167
Number of pages7
StatePublished - 1996
Externally publishedYes
EventProceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing - Philadelphia, PA, USA
Duration: May 22 1996May 24 1996

Other

OtherProceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing
CityPhiladelphia, PA, USA
Period5/22/965/24/96

Fingerprint

Linear equations
Linear algebra

ASJC Scopus subject areas

  • Software

Cite this

Allender, E., Beals, R., & Ogihara, M. (1996). Complexity of matrix rank and feasible systems of linear equations (extended abstract). In Conference Proceedings of the Annual ACM Symposium on Theory of Computing (pp. 161-167). ACM.

Complexity of matrix rank and feasible systems of linear equations (extended abstract). / Allender, Eric; Beals, Robert; Ogihara, Mitsunori.

Conference Proceedings of the Annual ACM Symposium on Theory of Computing. ACM, 1996. p. 161-167.

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

Allender, E, Beals, R & Ogihara, M 1996, Complexity of matrix rank and feasible systems of linear equations (extended abstract). in Conference Proceedings of the Annual ACM Symposium on Theory of Computing. ACM, pp. 161-167, Proceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing, Philadelphia, PA, USA, 5/22/96.
Allender E, Beals R, Ogihara M. Complexity of matrix rank and feasible systems of linear equations (extended abstract). In Conference Proceedings of the Annual ACM Symposium on Theory of Computing. ACM. 1996. p. 161-167
Allender, Eric ; Beals, Robert ; Ogihara, Mitsunori. / Complexity of matrix rank and feasible systems of linear equations (extended abstract). Conference Proceedings of the Annual ACM Symposium on Theory of Computing. ACM, 1996. pp. 161-167
@inproceedings{a71a4345bbfd4c06ab8f88c5a3654444,
title = "Complexity of matrix rank and feasible systems of linear equations (extended abstract)",
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-reducibility, 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.",
author = "Eric Allender and Robert Beals and Mitsunori Ogihara",
year = "1996",
language = "English (US)",
pages = "161--167",
booktitle = "Conference Proceedings of the Annual ACM Symposium on Theory of Computing",
publisher = "ACM",

}

TY - GEN

T1 - Complexity of matrix rank and feasible systems of linear equations (extended abstract)

AU - Allender, Eric

AU - Beals, Robert

AU - Ogihara, Mitsunori

PY - 1996

Y1 - 1996

N2 - 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-reducibility, 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.

AB - 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-reducibility, 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.

UR - http://www.scopus.com/inward/record.url?scp=0029723580&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029723580&partnerID=8YFLogxK

M3 - Conference contribution

SP - 161

EP - 167

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

PB - ACM

ER -