Hardness and approximation of minimum distortion embeddings

We show that the problem of computing a minimum distortion embedding of a given graph into a path remains NP-hard when the input graph is restricted to a bipartite, cobipartite, or split graph. This implies the NP-hardness of the problem also on chordal, cocomparability, and AT-free graphs. This problem is hard to approximate within a constant factor on arbitrary graphs. We give polynomial-time constant-factor approximation algorithms for split and cocomparability graphs. We conclude with some upper bounds for interval graphs and cographs, on which the computational complexity of the problem is open.