A Note on the Complexity of Computing the Number of Reachable Vertices in a Digraph

TL;DR

This paper proves that counting reachable vertices from each node in a directed graph cannot be done faster than a certain bound unless a major complexity hypothesis fails, even for acyclic graphs.

Contribution

It establishes a conditional lower bound on the computational complexity of counting reachable vertices in directed graphs, assuming the Strong Exponential Time Hypothesis.

Findings

No $ ilde{O}(|E|^{2- heta})$ algorithm exists unless SETH fails

The lower bound applies even to acyclic graphs

Supports the conjecture that the problem is inherently computationally hard

Abstract

In this work, we consider the following problem: given a digraph $G=(V,E)$, for each vertex $v$, we want to compute the number of vertices reachable from $v$. In other words, we want to compute the out-degree of each vertex in the transitive closure of $G$. We show that this problem is not solvable in time $\mathcal{O}\left(|E|^{2-\epsilon}\right)$ for any $\epsilon>0$, unless the Strong Exponential Time Hypothesis is false. This result still holds if $G$ is assumed to be acyclic.