The Circlet Inequalities: A New, Circulant-Based Facet-Defining Inequality for the TSP

TL;DR

This paper introduces the circlet inequalities, a new class of facet-defining inequalities for the TSP polytope that leverage circulant symmetry, bridging polyhedral TSP research with number theory, and demonstrating their strength and validity.

Contribution

The paper presents the first proof of the validity and facet-defining property of circlet inequalities, connecting circulant symmetry with TSP polyhedral theory and number-theoretic conjectures.

Findings

Circlet inequalities are valid and facet-defining for the TSP polytope.

They exhibit circulant symmetry and are generally stronger than existing inequalities.

They achieve the same worst-case strength as known circulant inequalities.

Abstract

Facet-defining inequalities of the symmetric Traveling Salesman Problem (TSP) polytope play a prominent role in both polyhedral TSP research and state-of-the-art TSP solvers. In this paper, we introduce a new class of facet-defining inequalities, the \emph{circlet inequalities}. These inequalities were first conjectured in Gutekunst and Williamson \cite{Gut19b} when studying Circulant TSP, and they provide a bridge between polyhedral TSP research and number-theoretic investigations of Hamiltonian cycles stemming from a conjecture due to Marco Buratti in 2017. The circlet inequalities exhibit circulant symmetry by placing the same weight on all edges of a given length; our main proof exploits this symmetry to prove the validity of the circlet inequalities. We then show that the circlet inequalities are facet-defining and compute their strength following Goemans \cite{Goe95b}; they achieve the same worst-case strength as the similarly circulant crown inequalities of Naddef and Rinaldi \cite{Nad92}, but are generally stronger.