Approximation Algorithms for Min-Distance Problems

Mina Dalirrooyfard; Virginia Vassilevska Williams; Nikhil Vyas; Nicole Wein; Yinzhan Xu; Yuancheng Yu

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arXiv · 1904.11606

Approximation Algorithms for Min-Distance Problems

Mina Dalirrooyfard·Virginia Vassilevska Williams·Nikhil Vyas·Nicole Wein·Yinzhan Xu·Yuancheng Yu

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Abstract

We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between $u$ and $v$ is the minimum of the shortest path distances from $u$ to $v$ and from $v$ to $u$. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help. By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we study can be computed exactly in $\tilde{O}(mn)$ time for directed graphs on $n$ vertices, $m$ edges and nonnegative edge weights. Furthermore, this time bound is tight under the Strong Exponential Time Hypothesis [Roditty-Vassilevska W. STOC 2013] so it is natural to study how well these parameters can be approximated in $O(mn^{1-\epsilon})$ time for constant $\epsilon>0$. Abboud, Vassilevska Williams, and Wang [SODA 2016] gave a polynomial factor approximation for Diameter and Radius, as well as a constant factor approximation for both problems in the special case where the graph is a DAG. We greatly improve upon these bounds by providing the first constant factor approximations for Diameter, Radius and the related Eccentricities problem in general graphs. Additionally, we provide a hierarchy of algorithms for Diameter that gives a time/accuracy trade-off.

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Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, Yuancheng Yu. 2019-04-25. Approximation Algorithms for Min-Distance Problems. https://arxiv.org/abs/1904.11606

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