Single source shortest paths in $H$-minor free graphs

TL;DR

This paper introduces a faster algorithm for solving the Single Source Shortest Paths problem in $H$-minor free graphs, improving previous time bounds and efficiently computing shortest paths or certifying non-$H$-minor freeness.

Contribution

It presents a new algorithm with improved runtime for SSSP in $H$-minor free graphs, advancing the state-of-the-art in graph algorithms.

Findings

Achieves a runtime of approximately O(n^{1.392} log L) for fixed $H$.

Provides shortest paths or certifies non-$H$-minor freeness.

Improves upon the previous O(n^{1.5} log L) algorithm.

Abstract

We present an algorithm for the Single Source Shortest Paths (SSSP) problem in \emph{$H$-minor free} graphs. For every fixed $H$, if $G$ is a graph with $n$ vertices having integer edge lengths and $s$ is a designated source vertex of $G$, the algorithm runs in $\tilde{O}(n^{\sqrt{11.5}-2} \log L) \le O(n^{1.392} \log L)$ time, where $L$ is the absolute value of the smallest edge length. The algorithm computes shortest paths and the distances from $s$ to all vertices of the graph, or else provides a certificate that $G$ is not $H$-minor free. Our result improves an earlier $O(n^{1.5} \log L)$ time algorithm for this problem, which follows from a general SSSP algorithm of Goldberg.