computational complexity

, Volume 19, Issue 1, pp 57–98 | Cite as

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

  • V. ArvindEmail author
  • T. C. Vijayaraghavan


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 Axb 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 LGapL/poly. As LCON is hard for LGapL 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.


Logspace counting classes abelian permutation groups linear congruences randomized computation 

Subject classification.

68Q15 15A03 15A06 20B05 


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Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesChennaiIndia
  2. 2.Chennai Mathematical InstituteChennaiIndia

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