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

TL;DR: A simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding (unweighted) graph metrics into tree metrics is presented, improving and simplifying the factor 100 and 27 algorithms.

Abstract: We present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding (unweighted) graph metrics into tree metrics (thus improving and simplifying the factor 100 and 27 algorithms of Badoiu et al. (2007) and Badoiu et al. (2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding graph metrics into outerplanar metrics. For this, we introduce a notion of metric relaxed minor and show that if G contains an α-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is ≥ α. Then, for H = K2,3 we present an algorithm which either finds an α-relaxed minor, or produces an O(α)-embedding into an outerplanar metric.