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arXiv · 2609.34600

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

Zhao Song
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Abstract

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]. In this paper, we obtain a randomized polynomial-time $(3/2-\varepsilon)$-approximation for every fixed $0<\varepsilon<\varepsilon_\star$, where $\varepsilon_\star>2.78621\cdot10^{-18}$, and consequently the subtour-elimination LP has integrality gap at most $3/2-\varepsilon_\star$. The classical worst-case integrality-gap lower bound is $4/3$ [Wil90].

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Zhao Song. 2026-09-28. A Sharper Explicit Bound on the Subtour-LP Integrality Gap for Metric TSP. https://arxiv.org/abs/2609.34600

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