The Discretizable Molecular Distance Geometry Problem

Given a weighted undirected graph $G=(V,E,d)$, the Molecular Distance Geometry Problem (MDGP) is that of finding a function $x:G\to \mathbb{R}^{3}$, where $||x(u)-x(v)||=d(u,v)$ for each $\{u,v\}\in E$. We show that under a few assumptions usually satisfied in proteins, the MDGP can be formulated as a search in a discrete space. We call this MDGP subclass the Discretizable MDGP (DMDGP). We show that the DMDGP is \textbf{NP}-complete and we propose an algorithm, called Branch-and-Prune (BP), which solves the DMDGP exactly. The BP algorithm performs exceptionally well in terms of solution accuracy and can find all solutions to any DMDGP instance. We successfully test the BP algorithm on several randomly generated instances.