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International audienceWe present a simple factor 6 algorithm for approximatingthe optimal multiplicative distortion of embedding (unweighted)graph metrics into  tree metrics  (thus improving and simplifyingthe factor 100 and 27 algorithms of B\v{a}doiu et al. (2007) andB\v{a}doiu et al. (2008)). We also present a constant factoralgorithm for approximating the optimal distortion of embeddinggraph metrics into outerplanar metrics. For this, we introduce anotion of metric relaxed minor and show that if $G$ contains an$\alpha$-metric relaxed $H$-minor, then the distortion of anyembedding of $G$ into any metric induced by a $H$-minor free graphis $\geq \alpha$. Then, for $H=K_{2,3}$, we present an algorithmwhich either finds an $\alpha$-relaxed minor, or produces an$O(\alpha)$-embedding into an outerplanar metric

Constant Approximation Algorithms for Embedding Graph Metrics into Trees and Outerplanar Graphs