Discretization Orders for Distance Geometry Problems
C. Lavor, J. Lee, A. Lee-St.John, L. Liberti, A. Mucherino, M. Sviridenko

Given a weighted, undirected simple graph G = (V,E,d) the Distance Geometry Problem (DGP) consists in determining an embedding x such that for each (i,j) in E ||xi - xj|| = d(i,j) . 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 K. We exhibit computational experiments on a set of proteins whose natural atomic order does not satisfy the order requirements, and compare our approach with some available continuous space searches.