The iBP algorithm for the discretizable molecular distance geometry problem with interval data Carlile Lavor 1 , Leo Liberti 2 , Antonio Mucherino 3 1 Dept. of Applied Math. (IMECC-UNICAMP), State University of Campinas, Campinas - SP, Brazil Email:clavor@ime.unicamp.br 2 LIX, ´ Ecole Polytechnique, F-91128 Palaiseau, France Email:liberti@lix.polytechnique.fr 3 CERFACS, Toulouse, France Email:mucherino@cerfacs.fr January 6, 2011 Abstract The Distance Geometry Problem in three dimensions consists in finding an embedding in R 3 of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the vertices. In this paper we discuss the case where some of the edge weights are subject to uncertainty within a given nonnegative interval. We show that a discretization is still possible and propose the iBP algorithm to solve the problem. The approach is validated by some computational experiments on a set of artificially generated instances. 1 Introduction We consider the problem of determining a Euclidean embedding of a simple weighted graph, the so- called Distance Geometry Problem (DGP). This problem has at least three important applications: finding the three-dimensional conformation (the coordinates of all the atoms) of a molecule from a subset of inter-atomic distances found using Nuclear Magnetic Resonance (NMR) [10, 16]; finding the position of wireless sensors given some of the distances (estimated by monitoring the power needing to communicate with each sensor’s neighbours) [4, 24]; and graph drawing (www.graphdrawing.org). In this paper we consider the application to finding the three-dimensional conformation of proteins. In this case, this problem is usually referred to as Molecular DGP (MDGP). Proteins are important molecules which perform several functions in living beings. If their three-dimensional conformations are discovered, they are able to reveal the specific function that each protein is supposed to perform. A web database named Protein Data Bank (PDB) [1] is collecting all the three-dimensional conformations of proteins that scientists in the world have been able to obtain. To date, quite a few conformations on the PDB have been obtained through NMR experiments, where the corresponding MDGP has been solved by general-purpose continuous approaches for global optimization. The meta-heuristic Simulated Annealing [7, 21] is employed in most of the cases. Let G =(V,E,d) be a nonnegatively weighted simple undirected graph representing an instance of the MDGP. Vertices of G correspond to the atoms forming the molecule, and edges indicate if the distance between the respective atoms is known or not. Since we focus our attention on proteins and on NMR experiments for obtaining estimates of inter-atomic distances, we are able to make the following assumptions, which will allow us to discretize the problem: