On the enumerability of the determinant and the rank

Alina Beygelzimer, Mitsunori Ogihara

Research output: Chapter in Book/Report/Conference proceedingChapter

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 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.

Original languageEnglish (US)
Title of host publicationIFIP Advances in Information and Communication Technology
Pages59-70
Number of pages12
Volume96
StatePublished - 2002
Externally publishedYes
EventIFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002 - Montreal, QC, Canada
Duration: Aug 25 2002Aug 30 2002

Publication series

NameIFIP Advances in Information and Communication Technology
Volume96
ISSN (Print)18684238

Other

OtherIFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002
CountryCanada
CityMontreal, QC
Period8/25/028/30/02

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ASJC Scopus subject areas

  • Information Systems and Management

Cite this

Beygelzimer, A., & Ogihara, M. (2002). On the enumerability of the determinant and the rank. In 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.

IFIP Advances in Information and Communication Technology. Vol. 96 2002. p. 59-70 (IFIP Advances in Information and Communication Technology; Vol. 96).

Research output: Chapter in Book/Report/Conference proceedingChapter

Beygelzimer, A & Ogihara, M 2002, On the enumerability of the determinant and the rank. in 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.
Beygelzimer A, Ogihara M. On the enumerability of the determinant and the rank. In IFIP Advances in Information and Communication Technology. Vol. 96. 2002. p. 59-70. (IFIP Advances in Information and Communication Technology).
Beygelzimer, Alina ; Ogihara, Mitsunori. / On the enumerability of the determinant and the rank. IFIP Advances in Information and Communication Technology. Vol. 96 2002. pp. 59-70 (IFIP Advances in Information and Communication Technology).
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