  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.
