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

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 $\geq \alpha$. 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.
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https://hal.archives-ouvertes.fr/hal-01194791
Contributor : Yann Vaxès <>
Submitted on : Monday, September 7, 2015 - 3:27:41 PM
Last modification on : Monday, March 4, 2019 - 2:04:14 PM

### Citation

Victor Chepoi, Feodor Dragan, Ilan Newman, Yuri Rabinovich, Yann Vaxès. Constant Approximation Algorithms for Embedding Graph Metrics into Trees and Outerplanar Graphs. Discrete and Computational Geometry, Springer Verlag, 2012, 47 (1), pp.187--214. ⟨10.1007/s00454-011-9386-0⟩. ⟨hal-01194791⟩

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