A linear time algorithm for the orbit problem over cyclic groups

TL;DR

This paper presents a linear-time algorithm for solving the orbit problem specifically over cyclic groups, which is a significant restriction of the general problem and important for symmetry reduction in model checking.

Contribution

The paper introduces the first linear-time algorithm for the orbit problem when the permutation group is cyclic, improving upon previous polynomial-time solutions.

Findings

The algorithm runs in linear time for cyclic groups.

It efficiently determines orbit equivalence under cyclic group actions.

This advances symmetry reduction techniques in model checking.

Abstract

The orbit problem is at the heart of symmetry reduction methods for model checking concurrent systems. It asks whether two given configurations in a concurrent system (represented as finite strings over some finite alphabet) are in the same orbit with respect to a given finite permutation group (represented by their generators) acting on this set of configurations by permuting indices. It is known that the problem is in general as hard as the graph isomorphism problem, whose precise complexity (whether it is solvable in polynomial-time) is a long-standing open problem. In this paper, we consider the restriction of the orbit problem when the permutation group is cyclic (i.e. generated by a single permutation), an important restriction of the problem. It is known that this subproblem is solvable in polynomial-time. Our main result is a linear-time algorithm for this subproblem.