A note on the relationship between the Graphical Traveling Salesman Polyhedron, the Symmetric Traveling Salesman Polytope, and the Metric Cone

TL;DR

This paper reveals a surprisingly simple geometric relationship between the Graphical Traveling Salesman Polyhedron, the Symmetric Traveling Salesman Polytope, and the metric cone, enhancing understanding of their interconnections.

Contribution

It demonstrates that the Graphical TSP Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric TSP Polytope and the polar of the metric cone, clarifying their relationship.

Findings

The relationship is almost trivial from known facts.

It helps understand the connection between these polyhedra.

The result is surprising despite its simplicity.

Abstract

In this short communication, we observe that the Graphical Traveling Salesman Polyhedron is the intersection of the positive orthant with the Minkowski sum of the Symmetric Traveling Salesman Polytope and the polar of the metric cone. This follows almost trivially from known facts. There are two reasons why we find this observation worth communicating none-the-less: It is very surprising; it helps to understand the relationship between these two important families of polyhedra.