Abstract

Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices in a graph. Computing the Diameter of a graph with m edges requires m2−o(1) time under the Strong Exponential Time Hypothesis (SETH), which can be prohibitive for very large graphs, so efficient approximation algorithms for Diameter are desired. There is a folklore algorithm that gives a 2-approximation for Diameter in O(m) time (where O notation suppresses logarithmic factors). Additionally, a line of work [SODA’96, STOC’13, SODA’14] concludes with a 3/2-approximation algorithm for Diameter in weighted directed graphs that runs in O(m3/2) time. For directed graphs, these are the only known approximation algorithms for Diameter. The 3/2-approximation algorithm is known to be tight under SETH: Roditty and Vassilevska W. [STOC’13] proved that under SETH any 3/2−e approximation algorithm for Diameter in undirected unweighted graphs requires m2−o(1) time, and then Backurs, Roditty, Segal, Vassilevska W., and Wein [STOC’18] and the follow-up work of Li proved that under SETH any 5/3−e approximation algorithm for Diameter in undirected unweighted graphs requires m3/2−o(1) time. Whether or not the folklore 2-approximation algorithm is tight, however, is unknown, and has been explicitly posed as an open problem in numerous papers. Towards this question, Bonnet recently proved that under SETH, any 7/4−e approximation requires m4/3−o(1), only for directed weighted graphs. We completely resolve this question for directed graphs by proving that the folklore 2-approximation algorithm is conditionally optimal. In doing so, we obtain a series of conditional lower bounds that together with prior work, give a complete time-accuracy trade-off that is tight with the three known algorithms for directed graphs. Specifically, we prove that under SETH for any δ>0, a (2k−1/k−δ)-approximation algorithm for Diameter on directed unweighted graphs requires mk/k−1−o(1) time.

Highlights

  • Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices in a graph i.e. maxu,v ∈V d (u, v), where V is the vertex set

  • We study the standard version of Diameter, which is one of the central problems in fine-grained complexity

  • Since quadratic time can be prohibitively slow on very large graphs, finding efficient approximation algorithms for Diameter is desirable

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Summary

INTRODUCTION

Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices in a graph i.e. maxu,v ∈V d (u, v), where V is the vertex set. For the related problem of Eccentricities, where the goal is to find the largest distance from every vertex in the graph, there is an analogous folklore algorithm and an analogous Main Question, which was resolved in [4] They gave an O (m) algorithm for Eccentricities that improves over the folklore algorithm, and showed that it is conditionally tight. Li [22] (in a earlier version of his paper) improved the unweighted undirected construction of [4] to match the weighted undirected construction of [4] That is, he showed that under SETH, any 5/3 − ε approximation algorithm for Diameter in undirected unweighted graphs requires m3/2−o (1) time. 7/4 and 2 for the optimal approximation factor for an O (m)-time algorithm for Diameter in directed weighted graphs, and a gap between 5/3 and 2 for undirected unweighted graphs

Our Results
Our Techniques
Organization
PRELIMINARIES
THE CONSTRUCTION
Vertex Set
Edge Set
NO INSTANCE OF k-OV IMPLIES DIAMETER AT MOST k
Fixed Paths
Variable Paths
NO Instance of 4-OV Implies Diameter at Most 4
YES Instance of 4-OV Implies Diameter at Least 7
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