TSP integrality gap via 2-edge-connected multisubgraph problem under coincident IP optima
Toshiaki Yamanaka
TL;DR
Problem: understanding the metric TSP integrality gap via connections to the 2ECM problem. Approach: a transfer principle demonstrates that when the 2ECM-IP optimum is a unique Hamiltonian cycle, approximation guarantees transfer directly to TSP and LP relaxations coincide. Contributions: (i) a formal transfer theorem; (ii) a general cut-margin stability framework certifying uniqueness and its stability under cost perturbations; (iii) an open problem to construct explicit instances where half-integral LP solutions coexist with a unique IP optimum; (iv) discussion of implications for a 4/3 bound in half-integral cases. Significance: provides a structured pathway to relate 2ECM techniques to TSP approximations and clarifies the role of LP relaxations in this transfer.
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 stability 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.