Lower Bounds for Symbolic Computation on Graphs: Strongly Connected Components, Liveness, Safety, and Diameter

TL;DR

This paper establishes linear lower bounds on the number of symbolic operations needed for fundamental graph problems like strongly connected components and diameter, highlighting the computational complexity in symbolic algorithms for reactive systems.

Contribution

It provides the first known lower bounds for key graph problems in symbolic computation and introduces an efficient approximation algorithm for graph diameter.

Findings

Lower bounds are linear in the number of vertices for several problems.

An approximation algorithm for diameter achieves a (1+ε)-approximation with (n  D) symbolic steps.

Refined analysis shows existing SCC algorithms are optimal in symbolic steps.

Abstract

A model of computation that is widely used in the formal analysis of reactive systems is symbolic algorithms. In this model the access to the input graph is restricted to consist of symbolic operations, which are expensive in comparison to the standard RAM operations. We give lower bounds on the number of symbolic operations for basic graph problems such as the computation of the strongly connected components and of the approximate diameter as well as for fundamental problems in model checking such as safety, liveness, and co-liveness. Our lower bounds are linear in the number of vertices of the graph, even for constant-diameter graphs. For none of these problems lower bounds on the number of symbolic operations were known before. The lower bounds show an interesting separation of these problems from the reachability problem, which can be solved with $O(D)$ symbolic operations, where $D$ is the diameter of the graph. Additionally we present an approximation algorithm for the graph diameter which requires $\tilde{O}(n \sqrt{D})$ symbolic steps to achieve a $(1+\epsilon)$-approximation for any constant $\epsilon > 0$. This compares to $O(n \cdot D)$ symbolic steps for the (naive) exact algorithm and $O(D)$ symbolic steps for a 2-approximation. Finally we also give a refined analysis of the strongly connected components algorithms of Gentilini et al., showing that it uses an optimal number of symbolic steps that is proportional to the sum of the diameters of the strongly connected components.