Discretization orders for distance geometry problems

Given a weighted, undirected simple graph G = (V, E, d) (where d: E → ℝ ₊), the distance geometry problem (DGP) is to determine an embedding x: V → ℝ ^K such that ∀{i, j} ∈ E {double pipe} X _i - x _j {double pipe} = d _ij. Although, in general, the DGP is solved using continuous methods, under certain conditions the search is reduced to a discrete set of points. We give one such condition as a particular order on V. We formalize the decision problem of determining whether such an order exists for a given graph and show that this problem is NP-complete in general and polynomial for fixed dimension K. We present results of computational experiments on a set of protein backbones whose natural atomic order does not satisfy the order requirements and compare our approach with some available continuous space searches.