Discretization orders for distance geometry problems

Carlile Lavor, Jon Lee, Audrey Lee-St. John, Leo Liberti, Antonio Mucherino, Maxim Sviridenko

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