Optimal Partial Discretization Orders for Discretizable Distance Geometry
Douglas S. Gonçalves, Antonio Mucherino

The Distance Geometry Problem (DGP) asks whether a simple weighted undirected graph G = (V,E,d) can be embedded in a given space so that the weights of the edges of G, when available, are the same as the distances between pairs of embedded vertices. The DGP can be discretized when some particular assumptions are satisfied, which are strongly dependent on the vertex ordering assigned to G. In this work, we focus on the problem of identifying optimal partial discretization orders for the DGP. The solutions to this problem are in fact vertex orders that allow for the discretization of the DGP. Moreover, these partial orders are optimal in the sense that they optimize, at each rank, a given set of objectives aimed to improve the structure of the search space after the discretization. This ordering problem is tackled from a theoretical point of view, and some practical experiences on sets of artificially generated instances, as well as on real-life instances, are provided.