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On Diameter Approximation in Directed Graphs

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

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Author Details

Amir Abboud
  • Weizmann Institute of Science, Rehovot, Israel
Mina Dalirrooyfard
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Ray Li
  • University of California Berkeley, CA, USA
Virginia Vassilevska Williams
  • Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Abboud, Amir: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme (grant agreement No 101078482). Additionally, Amir Abboud is supported by an Alon scholarship and a research grant from the Center for New Scientists at the Weizmann Institute of Science.
  • Dalirrooyfard, Mina: Partially supported by an Akamai Fellowship.
  • Li, Ray: Supported by the NSF Mathematical Sciences Postdoctoral Research Fellowships Program under Grant DMS-2203067, and a UC Berkeley Initiative for Computational Transformation award.
  • Vassilevska Williams, Virginia: Partially supported by the National Science Foundation Grant CCF-2129139.

Acknowledgements

We would like to thank Piotr Indyk, Karthik C.S., and the participants of the Fine-Grained Approximation Algorithms & Complexity Workshop (FG-APX 2019) at Bertinoro 2019 for many helpful discussions.

Cite As Get BibTex

Amir Abboud, Mina Dalirrooyfard, Ray Li, and Virginia Vassilevska Williams. On Diameter Approximation in Directed Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ESA.2023.2

Abstract

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 (max_{u,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 n^{1.5-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Graph algorithms analysis
Keywords
  • Diameter
  • Directed Graphs
  • Approximation Algorithms
  • Fine-grained complexity

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