Bicriteria approximation for minimum dilation graph augmentation

TL;DR

This paper presents a bicriteria approximation algorithm for adding a limited number of edges to a graph to minimize its dilation, improving upon previous linear-factor results and providing tight analysis under the Erdős girth conjecture.

Contribution

The paper introduces a novel bicriteria approximation algorithm with sublinear edge addition and controlled dilation, advancing the understanding of network augmentation.

Findings

Achieves a $(2 oot[r]{2} \, k^{1/r}, 2r)$-bicriteria approximation.

Runs in $O(n^3 \, ext{log} \, n)$ time for all $r \, ext{geq} \, 1$.

Analysis is tight under the Erdős girth conjecture.

Abstract

Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget $k$, which $k$ edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in $k$. Our main result is a $(2 \sqrt[r]{2} \ k^{1/r},2r)$-bicriteria approximation that runs in $O(n^3 \log n)$ time, for all $r \geq 1$. In other words, if $t^*$ is the minimum dilation after adding any $k$ edges to a graph, then our algorithm adds $O(k^{1+1/r})$ edges to the graph to obtain a dilation of $2rt^*$. Moreover, our analysis of the algorithm is tight under the Erd\H{o}s girth conjecture.