Improving the dilation of a metric graph by adding edges

TL;DR

This paper presents the first approximation algorithm for adding a limited number of edges to a metric graph to minimize its dilation, addressing a key open problem in graph augmentation.

Contribution

It introduces an $O(k)$-approximation algorithm for selecting edges to improve graph dilation with a given budget, a significant advancement over previous work.

Findings

Provides the first positive approximation algorithm for $k > 1$

Achieves an $O(k)$-approximation ratio

Runs in $O(n^3 \log n)$ time

Abstract

Most of the literature on spanners focuses on building the graph from scratch. This paper instead focuses on adding edges to improve an existing graph. A major open problem in this field is: given a graph embedded in a metric space, and a budget of $k$ edges, which $k$ edges do we add to produce a minimum-dilation graph? The special case where $k=1$ has been studied in the past, but no major breakthroughs have been made for $k > 1$. We provide the first positive result, an $O(k)$-approximation algorithm that runs in $O(n^3 \log n)$ time.