We describe a (1+ε)-approximation algorithm for finding the minimum distortion embedding of an n-point metric space X into the shortest path metric space of a weighted graph G with m vertices. The running time of our algorithm is parametrized by the values of the minimum distortion, δ opt , the spread, ∆, of the points of X, the treewidth, ω, of G, and the doubling dimension, λ, of G. In particular, our result implies a PTAS provided an X with polynomial spread, and the doubling dimension of G, the treewidth of G, and δ opt , are all constant. For example, if X has a polynomial spread and δ opt is a constant, we obtain PTAS's for embedding X into the following spaces: the line, a cycle, a bounded degree unweighted tree, and a bounded degree unweighted k-outer planar graph (for a constant k).
In a very recent work Nayyeri and Raichel REF obtained a (1 + ε)-approximation algorithm for finding the minimum-distortion embedding of an n-point metric space M into the shortest path metric space of a weighted graph H with m vertices.
A Treehouse with Custom Windows: Minimum Distortion Embeddings into Bounded Treewidth Graphs
SODA
Mathematics Computer Science