Distortion Is Fixed Parameter Tractable

Abstract

We study low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Let M = M(G) be the shortest path metric of an edge weighted graph G, with the vertex set V(G) and the edge set E(G), on n vertices. We give the first fixed parameter tractable algorithm that for an unweighted graph metric M and integer d either constructs an embedding of M into the line with distortion at most d, or concludes that no such embedding exists. Our algorithm requires O(nd ⁴(2d + 1)^2d) time which is a significant improvement over the best previous algorithm of Bădoiu et al. that runs in time O(n ^4d + 2 d ^O(1)). We find it surprising that this problem turns out to be fixed parameter tractable, because of its apparent similarity to the notoriously hard Bandwidth Minimization problem.

We extend our results on embedding unweighted graph metric into the line in two ways. First, we give an algorithm to construct small distortion embeddings of weighted graph metrics. The running time of our algorithm is O(n(dW)⁴(2d + 1)^2dW) where W is the largest edge weight of the input graph. To complement this result, we show that the exponential dependence on the maximum edge weight is unavoidable. In particular, we show that deciding whether a weighted graph metric M(G) with maximum weight W < |V(G)| can be embedded into the line with distortion at most d is NP-Complete for every fixed rational d ≥ 2. This rules out any possibility of an algorithm with running time O((nW)^h(d)) where h is a function of d alone. Secondly, we consider more general host metrics for which analogous results hold. In particular, we prove that for any tree T with maximum degree Δ, embedding M into a shortest path metric of T is fixed parameter tractable, parameterized by (Δ,d).