### Abstract

We investigate the complexity of enumerative approximation of two elementary problems in linear algebra, computing the rank and the determinant of a matrix. In particular, we show that if there exists an enumerator that, given a matrix, outputs a list of constantly many numbers, one of which is guaranteed to be the rank of the matrix, then it can be determined in AC^{0} (with oracle access to the enumerator) which of these numbers is the rank. Thus, for example, if the enumerator is an FL function, then the problem of computing the rank is in FL. The result holds for matrices over any commutative ring whose size grows at most polynomially with the size of the matrix. The existence of such an enumerator also implies a slightly stronger collapse of the exact counting logspace hierarchy. For the determinant function we establish the following two results: (1) If the determinant is poly-enumerable in logspace, then it can be computed exactly in FL. (2) For any prime p, if computing the determinant modulo p is (p - 1)-enumerable in FL, then computing the determinant modulo p can be done in FL. This gives a new perspective on the approximability of many elementary linear algebra problems equivalent to computing the rank or the determinant.

Original language | English (US) |
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Title of host publication | IFIP Advances in Information and Communication Technology |

Pages | 59-70 |

Number of pages | 12 |

Volume | 96 |

State | Published - 2002 |

Externally published | Yes |

Event | IFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002 - Montreal, QC, Canada Duration: Aug 25 2002 → Aug 30 2002 |

### Publication series

Name | IFIP Advances in Information and Communication Technology |
---|---|

Volume | 96 |

ISSN (Print) | 18684238 |

### Other

Other | IFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002 |
---|---|

Country | Canada |

City | Montreal, QC |

Period | 8/25/02 → 8/30/02 |

### Fingerprint

### ASJC Scopus subject areas

- Information Systems and Management

### Cite this

*IFIP Advances in Information and Communication Technology*(Vol. 96, pp. 59-70). (IFIP Advances in Information and Communication Technology; Vol. 96).

**On the enumerability of the determinant and the rank.** / Beygelzimer, Alina; Ogihara, Mitsunori.

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

*IFIP Advances in Information and Communication Technology.*vol. 96, IFIP Advances in Information and Communication Technology, vol. 96, pp. 59-70, IFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002, Montreal, QC, Canada, 8/25/02.

}

TY - CHAP

T1 - On the enumerability of the determinant and the rank

AU - Beygelzimer, Alina

AU - Ogihara, Mitsunori

PY - 2002

Y1 - 2002

N2 - We investigate the complexity of enumerative approximation of two elementary problems in linear algebra, computing the rank and the determinant of a matrix. In particular, we show that if there exists an enumerator that, given a matrix, outputs a list of constantly many numbers, one of which is guaranteed to be the rank of the matrix, then it can be determined in AC0 (with oracle access to the enumerator) which of these numbers is the rank. Thus, for example, if the enumerator is an FL function, then the problem of computing the rank is in FL. The result holds for matrices over any commutative ring whose size grows at most polynomially with the size of the matrix. The existence of such an enumerator also implies a slightly stronger collapse of the exact counting logspace hierarchy. For the determinant function we establish the following two results: (1) If the determinant is poly-enumerable in logspace, then it can be computed exactly in FL. (2) For any prime p, if computing the determinant modulo p is (p - 1)-enumerable in FL, then computing the determinant modulo p can be done in FL. This gives a new perspective on the approximability of many elementary linear algebra problems equivalent to computing the rank or the determinant.

AB - We investigate the complexity of enumerative approximation of two elementary problems in linear algebra, computing the rank and the determinant of a matrix. In particular, we show that if there exists an enumerator that, given a matrix, outputs a list of constantly many numbers, one of which is guaranteed to be the rank of the matrix, then it can be determined in AC0 (with oracle access to the enumerator) which of these numbers is the rank. Thus, for example, if the enumerator is an FL function, then the problem of computing the rank is in FL. The result holds for matrices over any commutative ring whose size grows at most polynomially with the size of the matrix. The existence of such an enumerator also implies a slightly stronger collapse of the exact counting logspace hierarchy. For the determinant function we establish the following two results: (1) If the determinant is poly-enumerable in logspace, then it can be computed exactly in FL. (2) For any prime p, if computing the determinant modulo p is (p - 1)-enumerable in FL, then computing the determinant modulo p can be done in FL. This gives a new perspective on the approximability of many elementary linear algebra problems equivalent to computing the rank or the determinant.

UR - http://www.scopus.com/inward/record.url?scp=84891052967&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84891052967&partnerID=8YFLogxK

M3 - Chapter

SN - 9781475752755

VL - 96

T3 - IFIP Advances in Information and Communication Technology

SP - 59

EP - 70

BT - IFIP Advances in Information and Communication Technology

ER -