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Given an integer dimension K and a simple, undirected graph G with positiveedge weights, the Distance Geometry Problem (DGP) aims to find a realizationfunction mapping each vertex to a coordinate in K-dimensional space such thatthe distance between pairs of vertex coordinates is equal to the correspondingedge weights in G. The so-called discretization assumptions reduce the searchspace of the realization to a finite discrete one which can be explored via thebranch-and-prune (BP) algorithm. Given a discretization vertex order in G, theBP algorithm constructs a binary tree where the nodes at a layer provide allpossible coordinates of the vertex corresponding to that layer. The focus ofthis paper is finding optimal BP trees for a class of Discretizable DGPs. Morespecifically, we aim to find a discretization vertex order in G that yields aBP tree with the least number of branches. We propose an integer programmingformulation and three constraint programming formulations that allsignificantly outperform the state-of-the-art cutting plane algorithm for thisproblem. Moreover, motivated by the difficulty in solving instances with alarge and low density input graph, we develop two hybrid decompositionalgorithms, strengthened by a set of valid inequalities, which further improvethe solvability of the problem.

Integer Programming, Constraint Programming, and Hybrid Decomposition Approaches to Discretizable Distance Geometry Problems