On the facial structure of Symmetric and Graphical Traveling Salesman Polyhedra

TL;DR

This paper investigates the relationship between the symmetric and graphical Traveling Salesman Polytopes, focusing on how valid inequalities can be transferred between them and revealing connections to polyhedral properties.

Contribution

It introduces a method for lifting inequalities from the symmetric to the graphical TSP polytope and uncovers a link between relaxation properties and ridge graph connectivity.

Findings

Valid inequalities can be 'lifted' from $S_n$ to $P_n$.

A connection between relaxations and ridge graph connectivity is established.

The study provides new insights into the polyhedral structure of TSP polytopes.

Abstract

The Symmetric Traveling Salesman Polytope $S_n$ for a fixed number $n$ of cities is a face of the corresponding Graphical Traveling Salesman Polyhedron $P_n$. This has been used to study facets of $S_n$ using $P_n$ as a tool. In this paper, we study the operation of "rotating" (or "lifting") valid inequalities for $S_n$ to obtain a valid inequalities for $P_n$. As an application, we describe a surprising relationship between (a) the parsimonious property of relaxations of the Symmetric Traveling Salesman Polytope and (b) a connectivity property of the ridge graph of the Graphical Traveling Salesman Polyhedron.