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 language | English (US) |
|---|---|
| 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 |
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ASJC Scopus subject areas
- Information Systems and Management
Cite this
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 proceeding › Chapter
}
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.
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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 -