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

## TL;DR

This paper introduces new integer and constraint programming methods, along with hybrid algorithms, to optimize branch-and-prune trees for discretizable Distance Geometry Problems, significantly improving solution efficiency.

## Contribution

It presents novel integer and constraint programming formulations and hybrid decomposition algorithms that outperform existing methods in solving discretizable DGPs.

## Key findings

- Proposed formulations outperform the cutting plane algorithm.
- Hybrid algorithms improve solvability for large, low-density graphs.
- Valid inequalities enhance the effectiveness of the decomposition methods.

## Abstract

Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in K-dimensional space such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes at a layer provide all possible coordinates of the vertex corresponding to that layer. The focus of this paper is finding optimal BP trees for a class of Discretizable DGPs. More specifically, we aim to find a discretization vertex order in G that yields a BP tree with the least number of branches. We propose an integer programming formulation and three constraint programming formulations that all significantly outperform the state-of-the-art cutting plane algorithm for this problem. Moreover, motivated by the difficulty in solving instances with a large and low density input graph, we develop two hybrid decomposition algorithms, strengthened by a set of valid inequalities, which further improve the solvability of the problem.