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.

翻译：本文研究在边权图中寻找至多经过 $h$ 条边的 $s$-$t$ 最短路径问题。贝尔曼-福特算法以 $O(hm)$ 时间复杂度求解该问题（其中 $m$ 为边数）。我们证明，在主流细粒度复杂度假设下，该运行时间在次多项式因子范围内达到最优。具体而言，在APSP假设下，即使针对具有非负边权的无向图，该问题也无法被加速求解。该下界对任意稠密度图及任意 $h \in O(\sqrt{m})$ 均成立。进一步地，在更强的Min-Plus卷积假设下，可消除 $h \in O(\sqrt{m})$ 的限制。换言之，$O(hm)$ 时间界在参数 $h$、$m$ 及节点数 $n$ 的完整参数空间中均为紧界。我们的下界结果可与近期针对负权单源最短路径问题（贝尔曼-福特算法的经典应用场景）提出的近线性时间算法形成鲜明对比。