# 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 ∈ O(√(m)). Moreover, under a stronger assumption, namely the Min-Plus Convolution Hypothesis, we can eliminate the restriction h ∈ O(√(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.

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