Solving distance geometry problem with inexact distances in integer plane

Abstract

Given the pairwise distances for a set of unknown points in a known metric space, the distance geometry problem (DGP) is to compute the point coordinates in conformation with the distance constraints. It is a well-known problem in the Euclidean space, has several variations, finds many applications, and so has been attempted by different researchers from time to time. However, to the best of our knowledge, it is not yet fully addressed to its merit, especially in the discrete space. Hence, in this paper we introduce a novel variant of DGP where the pairwise distance between every two unknown points is given a tolerance zone with the objective of finding the solution as a collection of integer points. The solution is based on characterization of different types of annulus intersection, their equivalence, and cardinality bounds of integer points. Necessary implementation details and useful heuristics make it attractive for practical applications in the discrete space.