Blazing a trail via matrix multiplications: A faster algorithm for non-shortest induced paths

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 $O\left(n^{18}\right)$ 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 $O\left(n^{4.75}\right)$-time algorithm. Our result improves the previous $O\left(n^{21}\right)$-time algorithm by Cook, Horsfield, Preissmann, Robin, Seymour, Sintiari, Trotignon, and Vušković [Journal of Combinatorial Theory, Series B, 2024] for recognizing graphs with all holes the same length, and reduces the running time to $O\left(n^{7.75}\right)$.