Bellman-Ford Is Optimal for Shortest Hop-Bounded Paths

Authors Tomasz Kociumaka , Adam Polak

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Author Details

Tomasz Kociumaka
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Adam Polak
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany


The second author would like to thank Danupon Nanongkai and Luca Trevisan for bringing his attention to the problem discussed in this paper, Alexandra Lassota - for useful feedback on an early draft of the manuscript, and Imbir - a ginger tabby, who supervised initial stages of this work.

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Tomasz Kociumaka and Adam Polak. Bellman-Ford Is Optimal for Shortest Hop-Bounded Paths. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 72:1-72:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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 nonnegative 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Fine-grained complexity
  • graph algorithms
  • lower bounds
  • shortest paths


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