On Saxe's theorems about the complexity of the Distance Geometry Problem

TL;DR

This paper revisits and clarifies Saxe's 1979 results on the complexity of the Distance Geometry Problem, providing detailed and more understandable proofs to enhance comprehension.

Contribution

It offers a detailed commentary and clearer proofs of Saxe's original theorems on the complexity of the Distance Geometry Problem.

Findings

Clarified the complexity status of the Distance Geometry Problem

Provided detailed proofs of Saxe's theorems

Enhanced understanding of the original results

Abstract

In 1979, James B.~Saxe published an extended summary on the complexity of the Distance Geometry Problem in the proceedings of the 17th Allerton Conference. Many of the proofs in his paper are sketches, and even the whole proofs do not have all the details. In this paper we provide a commentary to Saxe's results and hopefully more understandable versions thereof.