A Sharper Explicit Bound on the Subtour-LP Integrality Gap for Metric TSP

Karlin, Klein, and Oveis Gharan introduced a randomized better-than-\(3/2\) approximation algorithm for metric TSP [KKO21] and subsequently established the corresponding improvement in the integrality gap of the subtour-elimination LP [KKO22], with an explicit constant \(\varepsilon>1.00000\cdot10^{-36}\). Gurvits, Klein, and Leake subsequently improved the certified saving to \(2.18000\cdot10^{-34}\) [GKL24]. We further obtain a randomized polynomial-time $(3/2-\varepsilon)$-approximation for every fixed \(\varepsilon<\varepsilon_\star\), where \(\varepsilon_\star>2.05522\cdot10^{-30}\), and consequently the subtour-elimination LP has integrality gap at most \(3/2-\varepsilon_\star\).