The Circlet Inequalities: A New, Circulant-Based Facet-Defining Inequality for the TSP
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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 circlet inequalities. These inequalities were first conjectured in Gutekunst and Williamson <cit.> 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 <cit.>; they achieve the same worst-case strength as the similarly circulant crown inequalities of Naddef and Rinaldi <cit.>, but are generally stronger.
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