Cycle-based formulations in Distance Geometry

TL;DR

This paper introduces a novel cycle-based formulation for the distance geometry problem, focusing on cycle segment lengths rather than vertex positions, and compares its effectiveness to traditional edge-based models.

Contribution

The paper presents an innovative cycle-based modeling approach for the distance geometry problem, including an exact and a relaxed formulation, and evaluates their performance against existing methods.

Findings

Cycle-based formulations improve solution quality over edge-based models.

Edge-based models are faster but less accurate.

Cycle-based models are promising for protein conformation problems.

Abstract

The distance geometry problem asks to find a realization of a given simple edge-weighted graph in a Euclidean space of given dimension K, where the edges are realized as straight segments of lengths equal (or as close as possible) to the edge weights. The problem is often modelled as a mathematical programming formulation involving decision variables that determine the position of the vertices in the given Euclidean space. Solution algorithms are generally constructed using local or global nonlinear optimization techniques. We present a new modelling technique for this problem where, instead of deciding vertex positions, formulations decide the length of the segments representing the edges in each cycle in the graph, projected in every dimension. We propose an exact formulation and a relaxation based on a Eulerian cycle. We then compare computational results from protein conformation instances obtained with stochastic global optimization techniques on the new cycle-based formulation and on the existing edge-based formulation. While edge-based formulations take less time to reach termination, cycle-based formulations are generally better on solution quality measures.