# Approximation Algorithms for Min-Distance Problems

## TL;DR

This paper introduces the first constant factor approximation algorithms for Diameter, Radius, and Eccentricities in directed graphs under a novel distance measure, improving previous bounds and offering a trade-off between time and accuracy.

## Contribution

It presents the first constant factor approximation algorithms for key graph parameters in general directed graphs with a new distance measure, surpassing prior polynomial and DAG-specific bounds.

## Key findings

- Achieved constant factor approximations for Diameter, Radius, and Eccentricities.
- Developed a hierarchy of algorithms balancing time and approximation accuracy.
- Proved the tightness of exact computation time bounds under the Strong Exponential Time Hypothesis.

## 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.