度量TSP的Subtour-LP积分间隙的更紧显式上界
A Sharper Explicit Bound on the Subtour-LP Integrality Gap for Metric TSP
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中文总结 AI 辅助
本文针对度量TSP，改进了subtour-消去LP的积分间隙上界，获得随机多项式时间(3/2-ε)近似算法，显式常数ε_*>2.78621×10^-18，优于此前结果。
中文摘要 AI 辅助
Karlin、Klein和Oveis Gharan引入了度量TSP的随机化优于3/2近似算法[KKO21]，随后建立了subtour-消去LP积分间隙的相应改进[KKO22]，显式常数ε>1.00000×10^-36。Gurvits、Klein和Leake随后将认证节省改进至2.18000×10^-34[GKL24]。本文中，我们为每个固定的0<ε<ε_*获得随机多项式时间(3/2-ε)近似，其中ε_*>2.78621×10^-18，因此subtour-消去LP的积分间隙至多为3/2-ε_*。经典最坏情况积分间隙下界为4/3[Wil90]。
英文摘要
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].
