Is the Distance Geometry Problem in NP?

Given a weighted undirected graph G = (V,E,d) with d : E → Q+ and a positive integer K, the Distance Geometry Problem (DGP) asks to find an embedding x : V → R of G such that for each edge {i, j} we have ‖xi − x j‖ = di j. Saxe proved in 1979 that the DGP is NP-complete with K = 1 and doubted the applicability of the Turing machine model to the case with K > 1, because the certificates for YES instances might involve real numbers. This chapter is an account of an unfortunately failed attempt to prove that the DGP is in NP for K = 2. We hope that our failure will motivate further work on the question.