Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs

TL;DR

This paper introduces simple constant-factor approximation algorithms for embedding graph metrics into trees and outerplanar graphs, improving previous algorithms and providing new theoretical insights into metric embeddings.

Contribution

It presents a simplified factor 6 algorithm for tree embedding distortion and a novel constant factor algorithm for outerplanar embedding distortion, along with a new theoretical framework involving metric relaxed minors.

Findings

Improved approximation factor for tree metric embedding (factor 6).

New constant factor algorithm for outerplanar metric embedding.

Theoretical link between relaxed minors and embedding distortion.

Abstract

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of B\v{a}doiu, Indyk, and Sidiropoulos (2007) and B\v{a}doiu, Demaine, Hajiaghayi, Sidiropoulos, and Zadimoghaddam (2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of metric relaxed minor and show that if G contains an alpha-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is at meast alpha. Then, for H=K_{2,3}, we present an algorithm which either finds an alpha-relaxed minor, or produces an O(alpha)-embedding into an outerplanar metric.