Terminal Embeddings

TL;DR

This paper introduces terminal embeddings that preserve distances involving specific terminal points in a metric space, with distortion depending only on the number of terminals, and applies these embeddings to improve approximation algorithms for sparsest-cut problems.

Contribution

The paper develops new terminal embedding techniques with distortion bounds depending only on the number of terminals, and applies these to enhance sparsest-cut approximation algorithms.

Findings

Embeddings preserve terminal distances with distortion depending only on |K|.

Existence of embeddings that approximate all pairs while improving terminal pair distortion.

Application to improve sparsest-cut approximation ratio to O(\u221a{\log |K|})

Abstract

In this paper we study {\em terminal embeddings}, in which one is given a finite metric $(X,d_X)$ (or a graph $G=(V,E)$) and a subset $K \subseteq X$ of its points are designated as {\em terminals}. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve $\approx|K|\cdot |X|$ pairs, the distortion depends only on $|K|$, rather than on $|X|$. We also strengthen this notion, and consider embeddings that approximately preserve the distances between {\em all} pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to $X \times X$ and with respect to $K \times X$. Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, [ALN08] devised an $\tilde{O}(\sqrt{\log r})$-approximation algorithm for sparsest-cut instances with $r$ demands. Building on their framework, we provide an $\tilde{O}(\sqrt{\log |K|})$-approximation for sparsest-cut instances in which each demand is incident on one of the vertices of $K$ (aka, terminals). Since $|K| \le r$, our bound generalizes that of [ALN08].