A Coarse-Grained Representation for Discretizable Distance Geometry with Interval Data

We propose a coarse-grained representation for the solutions of discretizable instances of the Distance Geometry Problem (DGP). In several real-life applications, the distance information is not provided with high precision, but an approximation is rather given. We focus our attention on protein instances where inter-atomic distances can be either obtained from the chemical structure of the molecule (which are exact), or through experiments of Nuclear Magnetic Resonance (which are generally represented by real-valued intervals). The coarse-grained representation allows us to extend a previously proposed algorithm for the Discretizable DGP (DDGP), the branch-and-prune (BP) algorithm. In the standard BP, atomic positions are fixed to unique positions at every node of the search tree: we rather represent atomic positions by a pair consisting of a feasible region, together with a most-likely position for the atom in this region. While the feasible region is a constant during the search, the associated position can be refined by considering the new distance constraints that appear at further layers of the search tree. To perform the refinement task, we integrate the BP algorithm with a spectral projected gradient algorithm. Some preliminary computational experiments on artificially generated instances show that this new approach is quite promising to tackle real-life DGPs.