The Complexity of Bisimulation and Simulation on Finite Systems

TL;DR

This paper investigates the computational complexity of (bi)simulation problems on various finite graph classes, providing new complexity bounds and resolving an open problem in the field.

Contribution

It establishes the complexity of (bi)simulation on trees and related structures, solving an open problem and contrasting complexities for different graph classes.

Findings

(Bi)simulation on trees is complete for logarithmic space or NC^1.

If only one graph is a tree, (bi)simulation is in AC^1 or LogCFL.

Simulation is P-complete for graphs with bounded path-width.

Abstract

In this paper the computational complexity of the (bi)simulation problem over restricted graph classes is studied. For trees given as pointer structures or terms the (bi)simulation problem is complete for logarithmic space or NC$^1$, respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and S\'antha. Furthermore, if only one of the input graphs is required to be a tree, the bisimulation (simulation) problem is contained in AC$^1$ (LogCFL). In contrast, it is also shown that the simulation problem is P-complete already for graphs of bounded path-width.