Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems

Authors Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein

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Mina Dalirrooyfard
  • MIT, Cambridge, MA, USA
Virginia Vassilevska Williams
  • MIT, Cambridge, MA, USA
Nikhil Vyas
  • MIT, Cambridge, MA, USA
Nicole Wein
  • MIT, Cambridge, MA, USA


The authors would like to thank Arturs Backurs for discussions during the early stages of this work.

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Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, and Nicole Wein. Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 47:1-47:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important ST-variant considers two subsets S and T of the vertex set and lets the ST-diameter be the maximum distance between a node in S and a node in T, and the ST-radius be the minimum distance for a node of S to reach all nodes of T. The bichromatic variant is the special case in which S and T partition the vertex set. In this paper we present a comprehensive study of the approximability of ST and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis. For instance, for Bichromatic Diameter in undirected weighted graphs with m edges, we present an O~(m^{3/2}) time 5/3-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Shortest paths
  • approximation algorithms
  • fine-grained complexity
  • diameter
  • radius
  • eccentricities


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