Steiner Point Removal with Distortion $O(\log k)$

TL;DR

This paper improves the analysis of a ball-growing algorithm for the Steiner point removal problem, reducing the distortion bound from polylogarithmic to logarithmic in the number of terminals, thus achieving a tighter approximation.

Contribution

The paper provides a refined analysis of an existing algorithm, establishing an $O( ext{log }k)$ distortion bound for Steiner point removal, improving previous bounds.

Findings

Distortion bound improved to $O( ext{log }k)$

Analysis refines previous bounds of $O( ext{log}^2 k)$ and $O( ext{log}^5 k)$

Algorithm achieves tighter distance preservation in graph minors

Abstract

In the Steiner point removal (SPR) problem, we are given a weighted graph $G=(V,E)$ and a set of terminals $K\subset V$ of size $k$. The objective is to find a minor $M$ of $G$ with only the terminals as its vertex set, such that the distance between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer and Nguyen [KKN15] used a ball-growing algorithm with exponential distributions to show that the distortion is at most $O(\log^5 k)$. Cheung [Che17] improved the analysis of the same algorithm, bounding the distortion by $O(\log^2 k)$. We improve the analysis of this ball-growing algorithm even further, bounding the distortion by $O(\log k)$.