  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.
