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

TL;DR

This paper explores the relationship between the integrality gap of the metric TSP LP relaxation and the 2-edge-connected multisubgraph problem, introducing a transfer principle and stability framework to analyze unique optima and potential bounds.

Contribution

It introduces a transfer principle linking 2ECM solutions to TSP approximations and develops a stability framework to certify unique optima under perturbations.

Findings

If instances with unique Hamiltonian cycle and half-integral LP solutions exist, the TSP integrality gap is at most 4/3.

A transfer principle allows approximation results for 2ECM to imply results for TSP.

Constructing such instances remains an open problem.

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 $\alpha$-approximation algorithm for 2ECM that outputs a Hamiltonian cycle yields an $\alpha$-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.