Bellman-Ford in Almost-Linear Time for Dense Graphs

TL;DR

This paper presents a nearly linear time algorithm for solving the single-source shortest paths problem in dense graphs with possibly negative edge weights, improving computational efficiency significantly.

Contribution

It refines the shortcutting technique to achieve an almost-linear time complexity for dense graphs, advancing previous methods.

Findings

Achieves $n^{2+o(1)}$ time complexity for dense graphs

Handles graphs with negative edge weights

Improves upon previous shortest path algorithms

Abstract

We consider the single-source shortest paths problem on a directed graph with real-valued (possibly negative) edge weights and solve this problem in $n^{2+o(1)}$ time by refining the shortcutting procedure introduced in Li, Li, Rao, and Zhang (2026).