Multi-Source Reachability in Near-Optimal Time

2026-06-24

AI Summary

This work addresses the multi-source reachability problem in dense directed graphs with up to $n^\sigma$ sources, a setting where existing algorithms suffer from super-quadratic time bottlenecks. The paper presents the first near-optimal deterministic algorithm for this problem, leveraging rectangular matrix multiplication to achieve a running time of $\tilde{O}(n^{\omega(\sigma)})$. This result significantly improves upon the current best randomized algorithm by Elkin and Trehan (2024), whose complexity is $n^{1 + \frac{2}{3}\omega(\sigma)}$, thereby reducing the exponent to $\omega(\sigma)$. Notably, for $\sigma \leq 0.32$, the proposed method attains near-linear time performance, breaking through the previous reliance on randomization and offering a deterministic alternative with superior asymptotic guarantees.

Abstract

The multi-source reachability problem asks to compute the reachable sets from a given subset of source vertices. For $n$-vertex digraphs $G=(V,E)$ and a subset of sources $S \subseteq V$ with $|S|=n^σ$ for some $σ\in [0,1]$, we present a near-optimal deterministic algorithm that solves this problem in $\tilde{O}(n^{ω(σ)})$ time, where $ω(σ)$ is the rectangular matrix multiplication exponent for multiplying an $n^σ\times n$ matrix by an $n \times n$ matrix. For dense graphs, this yields reachability from up to $n^{0.32}$ sources in near-linear time, breaking the super-quadratic time barrier and improving over the state-of-the-art $n^{1+2/3ω(σ)}$-time randomized algorithm of Elkin and Trehan [arXiv:2401.05628, 2024].

Problem

• multi-source reachability
• directed graphs
• reachable sets
• graph algorithms
• near-optimal time

Innovation

• multi-source reachability
• rectangular matrix multiplication
• near-optimal algorithm
• deterministic algorithm
• graph algorithms