# The Hamiltonian Circuit Polytope

## TL;DR

This paper investigates the structure of the Hamiltonian circuit polytope, providing new tools and hierarchies of facet-defining inequalities to better understand and solve sequencing problems like the traveling salesman problem.

## Contribution

It introduces a novel approach based on variable index structure for identifying facets and develops a hierarchy of facet-defining inequalities with polynomial-time separation algorithms.

## Key findings

- Established the dimension of the Hamiltonian circuit polytope
- Developed tools for identifying facets using variable index structure
- Created a hierarchy of facet-defining inequalities with efficient algorithms

## Abstract

The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these tools to derive several families of facets. The tools include necessary and sufficient conditions for an inequality to be facet defining, and an algorithm for generating all undominated circuits. We use a novel approach to identifying families of facet-defining inequalities, based on the structure of variable indices rather than on subgraphs such as combs or subtours. This leads to our main result, a hierarchy of families of facet-defining inequalities and polynomial-time separation algorithms for them.