Constant Approximation of Min-Distances in Near-Linear Time

Abstract

In a weighed directed graph $G=(V, E, \omega)$ with m edges and n vertices, we are interested in its basic graph parameters such as diameter, radius and eccentricities, under the nonstandard measure of min-distance which is defined for every pair of vertices $u, v \in V$ as the minimum of the shortest path distances from u to v and from v to u. Similar to standard shortest paths distances, computing graph parameters exactly in terms of min-distances essentially requires $\tilde{\Omega}(m n)$ time under plausible hardness conjectures <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> . Hence, for faster running time complexities we have to tolerate approximations. Abboud, Vassilevska Williams and Wang [SODA 2016] were the first to study min-distance problems, and they obtained constant factor approximation algorithms in acyclic graphs, with running time $\tilde{O}(m)$ and $\tilde{O}(m \sqrt{n})$ for diameter and radius, respectively. The time complexity of radius in acyclic graphs was recently improved to $\tilde{O}(m)$ by Dalirrooyfard and Kaufmann [ICALP 2021], but at the cost of an $O(\log n)$ approximation ratio. For general graphs, the authors of [DWV+, ICALP 2019] gave the first constant factor approximation algorithm for diameter, radius and eccentricities which runs in time $\tilde{O}(m \sqrt{n})$; besides, for the diameter problem, the running time can be improved to $\tilde{O}(m)$ while blowing up the approximation ratio to $O(\log n)$. A natural question is whether constant approximation and near-linear time can be achieved simultaneously for diameter, radius and eccentricities; so far this is only possible for diameter in the restricted setting of acyclic graphs. In this paper, we answer this question in the affirmative by presenting near-linear time algorithms for all three parameters in general graphs. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> As usual, the $\tilde{O}(\cdot)$ notation hides poly-logarithmic factors in n