Constant Approximation of Min-Distances in Near-Linear Time

In a weighed directed graph G = (V, E, ?) 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 ? 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 O(m n) time under plausible hardness conjectures 1. 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 O(m) and O(m vn) for diameter and radius, respectively. The time complexity of radius in acyclic graphs was recently improved to 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 O(m vn); besides, for the diameter problem, the running time can be improved to 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.