Classifying Problems on Linear Congruences and Abelian Permutation Groups Using Logspace Counting Classes

Abstract.

In this paper we classify the complexity of several problems based on Abelian permutation groups and linear congruences using logspace counting classes. The problems we consider were defined by McKenzie & Cook (1987).

Central to our study is the problem LCON: given as input (A, b, q), where $A \in {\mathbb{Z}}^{m \times n}$ and b $\in {\mathbb{Z}}^m$, the problem is to determine if Ax = b is a feasible system of linear equations over ${\mathbb{Z}}_q$. We assume that q is given by its prime factorization $q = p^{e_1}_{1} p^{e_2}_{2} \cdot \cdot \cdot p^{e_k}_{k}$, such that each $p^{e_i}_i$ is tiny (i.e. given in unary). We give a randomized NC² algorithm for LCON. More precisely, LCON is in the nonuniform class L^GapL/poly. As LCON is hard for L^GapL we get a fairly tight characterization of LCON in terms of logspace counting classes. We derive the same upper bound for computing a basis for the nullspace of a linear map from ${\mathbb{Z}}^n_q$ to ${\mathbb{Z}}^m_q$. A number of Abelian permutation group problems studied in McKenzie & Cook (1987) turn out to be logspace Turing equivalent to these linear-algebraic problems. Consequently, the upper and lower bounds also carry over to these problems.