Parameterized Low-distortion Embeddings - Graph metrics into lines and trees

We revisit the issue of low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Low-distortion embeddings of a metric space into the line, or into some other “simple” metric space, (that is, a mapping that preserves distances, up to some small multiplicative factor called the distortion) have many applications in computer science. Let M = M(G) be the shortest path metric of an unweighted graph G = (V,E) on n vertices. We describe algorithms for the problem of finding a low distortion non-contracting embedding of M into line and tree metrics. • Our first result is that the problem of embedding M into the line, parameterized by the distortion d, is fixed parameter tractable (FPT). We describe an algorithm that on input (G, d) either constructs an embedding of M(G) into the line with distortion at most d, or concludes that no such embedding exists. The running time of our algorithm is O(nd(2d + 1)) which is linear for every fixed d and polynomial for d = O(lg n/ lg lg n). This is a significant improvement over the best previous algorithm, of Bădoiu et al. [SODA 2005] that has a running time of O(nd). • We generalize the result on embedding into the line by proving that for any tree T with maximum degree ∆, embedding of M into a shortest path metric of T is FPT, parameterized by (∆, d). This result can also be viewed as a generalization (albeit with a worse running time) of the previous FPT algorithm due to Kenyon, Rabani and Sinclair [STOC 2004] that was limited to the situation where |G| = |T |. University of Newcastle, Newcastle, Australia. {michael.fellows,frances.rosamond}@newcastle.edu.au Department of Informatics, University of Bergen, Bergen, Norway. {fedor.fomin,daniel.lokshtanov,saket.saurabh}@ii.uib.no Dept. of Computer Science, University of Iceland, Iceland. [email protected]