TSP integrality gap via 2-edge-connected multisubgraph problem under coincident IP optima

Abstract: Determining the integrality gap of the linear programming (LP) relaxation of the metric traveling salesman problem (TSP) remains a long-standing open problem. We introduce a transfer principle: when the integer optimum of the 2-edge-connected multisubgraph problem (2ECM) is a unique Hamiltonian cycle $T$, any $α$-approximation algorithm for 2ECM that outputs a Hamiltonian cycle yields an $α$-approximation for TSP. We further develop a cut-margin uniqueness framework that certifies $T$ as the unique integer optimum for both problems and is stable under $\ell_\infty$-bounded perturbations. We show that, if instances exist where the 2ECM has both a unique Hamiltonian cycle integer optimum and a half-integral LP solution, then the TSP integrality gap is at most 4/3 by the algorithm of Boyd et al. (SIAM Journal on Discrete Mathematics 36:1730--1747, 2022). Constructing such instances remains an open problem.