Distance geometry with and without the graph

TL;DR

This paper surveys the theoretical, algorithmic, and computational aspects of distance geometry problems, focusing on the differences in solvability and methods when adjacency information is available or not.

Contribution

It provides a comprehensive overview of distance geometry problems, highlighting the challenges and solutions with and without adjacency data in mathematical programming.

Findings

Mathematical programming effectively solves large-scale problems with adjacency.

Problems without adjacency are significantly more challenging to solve.

The survey identifies key open questions and future research directions.

Abstract

We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can solve large-scale distance geometry problems with adjacencies, they are severely challenged in the absence thereof.