Discretization orders for distance geometry problems

Given a weighted, undirected simple graph G = (V, E, d) (where $d:E\to {ℝ}_{+}$ ), the distance geometry problem (DGP) is to determine an embedding $x:V\to {ℝ}^K$ such that $\forall \{i,j\}\in E\parallel x_i‐x_j\parallel =d_{\mathrm{ij}}$ . 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.