Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-shortest Induced Paths

TL;DR

This paper introduces a significantly faster algorithm for finding non-shortest induced paths (trails) between vertices in a graph, reducing the complexity from polynomial to matrix multiplication time.

Contribution

The paper presents a novel algorithm that leverages matrix multiplication to efficiently find non-shortest induced paths, improving the previous polynomial-time approach.

Findings

Reduced the algorithm's running time to approximately O(n^{4.75})

Utilized Boolean matrix multiplication to achieve speedup

Provided a practical approach for detecting non-shortest induced paths

Abstract

For vertices $u$ and $v$ of an $n$-vertex graph $G$, a $uv$-trail of $G$ is an induced $uv$-path of $G$ that is not a shortest $uv$-path of $G$. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in $O(n^{18})$ time, to either output a $uv$-trail of $G$ or ensure that $G$ admits no $uv$-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of $n^2\times n^2$ Boolean matrices, leading to a largely improved $O(n^{4.75})$-time algorithm.