TY - GEN

T1 - On the enumerability of the determinant and the rank

AU - Beygelzimer, Alina

AU - Ogihara, Mitsunori

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

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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U2 - 10.1007/978-0-387-35608-2_6

DO - 10.1007/978-0-387-35608-2_6

M3 - Conference contribution

AN - SCOPUS:84891052967

SN - 9781475752755

T3 - IFIP Advances in Information and Communication Technology

SP - 59

EP - 70

BT - Foundations of Information Technology in the Era of Network and Mobile Computing - IFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP Int. Conference on Theoretical Computer Science (TCS 2002)

PB - Springer New York LLC

T2 - IFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, TCS 2002

Y2 - 25 August 2002 through 30 August 2002

ER -