Improving the average dilation of a metric graph by adding edges

TL;DR

This paper addresses reducing the average dilation of a metric graph by adding a limited number of edges, providing an approximation algorithm that matches previous bounds for the maximum dilation problem.

Contribution

It introduces an $O(k)$ approximation algorithm for minimizing average dilation with edge additions, extending prior work on maximum dilation.

Findings

Provides an $O(k)$ approximation algorithm for average dilation reduction.

Matches the approximation ratio of prior maximum dilation algorithms.

Offers a new approach for graph augmentation to improve average pairwise distances.

Abstract

For a graph $G$ spanning a metric space, the dilation of a pair of points is the ratio of their distance in the shortest path graph metric to their distance in the metric space. Given a graph $G$ and a budget $k$, a classic problem is to augment $G$ with $k$ additional edges to reduce the maximum dilation. In this note, we consider a variant of this problem where the goal is to reduce the average dilation for pairs of points in $G$. We provide an $O(k)$ approximation algorithm for this problem, matching the approximation ratio given by prior work for the maximum dilation variant.