On Diameter Approximation in Directed Graphs

Amir Abboud, Mina Dalirrooyfard, Ray Li, Virginia Vassilevska Williams

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds. In directed graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since d(u, v) may not be the same as d(v, u), there are multiple ways to define the problem, the two most natural being the (one-way) diameter (max(u,v) d(u, v)) and the roundtrip diameter (maxu,v d(u, v) + d(v, u)). In this paper we make progress on the outstanding open question for each of them. We design the first algorithm for diameter in sparse directed graphs to achieve n1.5−ε time with an approximation factor better than 2. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication. We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a 1.5-approximation in subquadratic time would refute the All-Nodes k-Cycle hypothesis, and any (2 − ε)-approximation would imply a breakthrough algorithm for approximate ℓ-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH.

Original languageEnglish
Title of host publication31st Annual European Symposium on Algorithms, ESA 2023
EditorsInge Li Gortz, Martin Farach-Colton, Simon J. Puglisi, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772952
StatePublished - Sep 2023
Event31st Annual European Symposium on Algorithms, ESA 2023 - Amsterdam, Netherlands
Duration: 4 Sep 20236 Sep 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference31st Annual European Symposium on Algorithms, ESA 2023

All Science Journal Classification (ASJC) codes

  • Software


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