### 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 NC^{1}-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 language | English (US) |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Publisher | ACM |

Pages | 161-167 |

Number of pages | 7 |

State | Published - 1996 |

Externally published | Yes |

Event | Proceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing - Philadelphia, PA, USA Duration: May 22 1996 → May 24 1996 |

### Other

Other | Proceedings of the 1996 28th Annual ACM Symposium on the Theory of Computing |
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City | Philadelphia, PA, USA |

Period | 5/22/96 → 5/24/96 |

### Fingerprint

### ASJC Scopus subject areas

- Software

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

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.

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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 -