We give simply exponential lower bounds on the probabilities of a given strongly Rayleigh distribution, depending only on its expectation. This resolves a weak version of a problem left open by Karlin-Klein-Oveis Gharan in their recent breakthrough work on metric TSP, and this resolution leads to a minor improvement of their approximation factor for metric TSP. Our results also allow for a more streamlined analysis of the algorithm. To achieve these new bounds, we build upon the work of Gurvits-Leake on the use of the productization technique for bounding the capacity of a real stable polynomial. This technique allows one to reduce certain inequalities for real stable polynomials to products of affine linear forms, which have an underlying matrix structure. In this paper, we push this technique further by characterizing the worst-case polynomials via bipartitioned forests. This rigid combinatorial structure yields a clean induction argument, which implies our stronger bounds. In general, we believe the results of this paper will lead to further improvement and simplification of the analysis of various combinatorial and probabilistic bounds and algorithms.

翻译：我们给出了给定强瑞利分布概率的简单指数下界，该下界仅依赖于其期望。这解决了Karlin-Klein-Oveis Gharan在度量旅行商问题近期突破性工作中遗留的一个弱版本问题，并因此对其度量旅行商问题的近似因子实现了微小改进。我们的结果还使得算法分析更为简洁。为获得这些新界，我们借鉴了Gurvits-Leake在利用乘积化技术界定实稳定多项式容量方面的研究成果。该技术允许将实稳定多项式的某些不等式简化为具有底层矩阵结构的仿射线性形式乘积。在本文中，我们通过二分森林刻画最坏情形多项式，进一步推进了该技术。这种刚性的组合结构提供了清晰的归纳论证，从而导出我们更强的界。总体而言，我们相信本文的结果将促进各类组合与概率界及算法分析的进一步改进与简化。