The (Non)Enumerability of the Determinant and the Rank

Abstract.

We investigate the complexity of enumerative approximation of two fundamental problems in linear algebra, computing the rank and the determinant of a matrix. We show that both are as hard to approximate (in the enumerative sense) as to compute exactly. In particular, if there exists an enumerator that, given a matrix over some finite field, 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. 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 (i.e., if one could eliminate a single possible value for the determinant modulo p ), then the determinant modulo p can be computed in FL. Because there is a close connection between these two functions and logspace counting classes, we hope that our results can give a better understanding of the power of counting in logspace, and the relationships among the complexity classes sandwiched between NL and uniform TC¹ .