Bellman-Ford is optimal for shortest hop-bounded paths
This paper is about the problem of finding a shortest $s$-$t$ path using at most $h$ edges in edge-weighted graphs. The Bellman--Ford algorithm solves this problem in $O(hm)$ time, where $m$ is the number of edges. We show that this running time is optimal, up to subpolynomial factors, under popular fine-grained complexity assumptions. More specifically, we show that under the APSP Hypothesis the problem cannot be solved faster already in undirected graphs with non-negative edge weights. This lower bound holds even restricted to graphs of arbitrary density and for arbitrary $h \in O(\sqrt{m})$. Moreover, under a stronger assumption, namely the Min-Plus Convolution Hypothesis, we can eliminate the restriction $h \in O(\sqrt{m})$. In other words, the $O(hm)$ bound is tight for the entire space of parameters $h$, $m$, and $n$, where $n$ is the number of nodes. Our lower bounds can be contrasted with the recent near-linear time algorithm for the negative-weight Single-Source Shortest Paths problem, which is the textbook application of the Bellman--Ford algorithm.