Feasibility check for the distance geometry problem: an application to molecular conformations

Abstract

The distance geometry problem (DGP) consists in finding an embedding in a metric space of a given weighted undirected graph such that for each edge in the graph, the corresponding distance in the embedding belongs to a given distance interval. We discuss the relationship between the existence of a graph embedding in a Euclidean space and the existence of a graph embedding in a lattice. Different approaches, including two integer programming (IP) models and a constraint programming (CP) approach, are presented to test the feasibility of the DGP. The two IP models are improved with the inclusion of valid inequalities, and the CP approach is improved using an algorithm to perform a domain reduction. The main motivation for this work is to derive new pruning devices within branch-and-prune algorithms for instances occurring in real applications related to determination of molecular conformations, which is a particular case of the DGP. A computational study based on a set of small-sized instances from molecular conformations is reported. This study compares the running times of the different approaches to check feasibility.

Minimal NMR distance information for rigidity of protein graphs

“…Although atoms Hαi-1, H i are in consecutive amino acids, there is just one torsion angle (the one defined by { N i −1 , Cαi-1, C i −1 , N i }) related to the position of H i , because the peptide plane “constrains” the torsion angle defined by { Cαi-1, C i −1 , N i −1 , Cαi} to be π radians. In the worst case, supposing that the distance d(Hαi-1,Hi) is not available, we can use “implicit” information associated with the fact that the distance was not detected [1] or some estimations given in [67].…”

On distance graph coloring problems

“…In this section, we discuss feasibility conditions related to our proposed EQ‐CDGP problems. This kind of study has also been applied to other related DGP (Agra et al., 2017). Clearly, the problems involving inequality constraints are always feasible.…”