Algorithms for low-distortion embeddings into arbitrary 1-dimensional spaces

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a “simpler” metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, into the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs G, H and integer c, is it possible to embed G with distortion c into a graph homeomorphic to H? Then embedding into the line is the special case H = K2, and embedding into the cycle is the case H = K3, where Kk denotes the complete graph on k vertices. For this problem we give • an approximation algorithm, which in time f(H) · poly(n), for some function f , either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion poly(c); • an exact algorithm, which in time f ′(H, c) · poly(n), for some function f ′, either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion c. Prior to our work, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.