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

• 成果类型: Article
• 署名作者: Gutekunst, Samuel C.; Williamson, David P.
• 署名单位: Bucknell University; Bucknell University; Cornell University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1265
• 发表日期: 2023
• 页码: 393-418
• 关键词: traveling-salesman problem

摘要

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 [Gutekunst SC, Williamson DP (2019) Characterizing the integrality gap of the subtour LP for the circulant traveling salesman problem. SIAM J. Discrete Math. 33(4):2452-2478] when studying the circulant TSP, and they provide a bridge between polyhedral TSP research and number-theoretic investigations of Hamiltonian cycles stemming from a conjecture from Marco Buratti in 2007. 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 [Goemans MX (1995) Worst-case comparison of valid inequalities for the TSP. Math. Programming 69:335-349]; they achieve the same worst case strength as the similarly circulant crown inequalities of Naddef and Rinaldi [Naddef D, Rinaldi G (1992) The crown inequalities for the symmetric traveling salesman polytope. Math. Oper. Res. 17(2):308-326] but are generally stronger.