## 1. Introduction

We present a discrete formulation and a very fast and accurate solution method for a subclass of instances of the Molecular Distance Geometry Problem (MDGP) (Moré and Wu, 1997, 1999; Crippen and Havel, 1988; Dong and Wu, 2002; Hendrickson, 1995). The MDGP is related to the determination of the tridimensional structure of a molecule based on knowledge of some distances between pairs of atoms. The tridimensional structure is very important because it is associated to the physical and chemical properties of the molecule.

The MDGP can be seen as finding a distance-preserving immersion in of a given undirected weighted graph *G*=(*V*, *E*, *d* ), so it can be very naturally cast as a continuous search problem. Under three additional assumptions that are satisfied by most proteins (a very interesting and rich class of molecules), we transform the MDGP to a discrete search problem. The assumptions are:

- 1covalent bond lengths and angles are known;
- 2the molecule has the shape of a protein backbone, i.e. it is a sequence of
*n*atoms such that there is a covalent bond between every pair of consecutive atoms; - 3all distances between atoms separated by three covalent bonds are known;
- 4no bond angle is equal to
*k*π, for .

Naturally, distances between atoms separated by two covalent bonds can be easily calculated from the covalent bond lengths and bond angles. We remark that if Assumption 1 above is strengthened to require knowledge of the distances between atoms separated by *four* covalent bonds, then the problem becomes polynomially solvable (Dong and Wu, 2002; Eren et al., 2004). Further considerations on the complexity of the MDGP problem under assumption 1 are given in Lavor et al. (2006).

It should be noted that we refer to the MDGP as a precisely formalized decision problem, and not as a practical chemical problem. We therefore make three assumptions that in real life may be easily challenged: (a) a subset of exact (as opposed to approximate) distances is given as part of the input; (b) no measurement errors occur; (c) the optimal three-dimensional (3D) embedding of the graph is not influenced by a potential energy minimization term on the objective function. We refer to Klepeis et al. (1999) for a more realistic formulation.

In Section 2, we show a discrete formulation for the problem. In Section 3, we describe the Branch-and-Prune algorithm, which will be applied to the solution of the MDGP. The computational results are discussed in Section 4. Section 5 concludes the paper.