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

International audienceIn this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embeddinga 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 $\geq \alpha$. 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