Smoothed analysis is a method for analyzing the performance of algorithms, used especially for those algorithms whose running time in practice is significantly better than what can be proven through worst-case analysis. Spielman and Teng (STOC '01) introduced the smoothed analysis framework of algorithm analysis and applied it to the simplex method. Given an arbitrary linear program with $d$ variables and $n$ inequality constraints, Spielman and Teng proved that the simplex method runs in time $O(σ^{-30} d^{55} n^{86})$, where $σ> 0$ is the standard deviation of Gaussian distributed noise added to the original LP data. Spielman and Teng's result was simplified and strengthened over a series of works, with the current strongest upper bound being $O(σ^{-3/2} d^{13/4} \log(n)^{7/4})$ pivot steps due to Huiberts, Lee and Zhang (STOC '23). We prove that there exists a simplex method whose smoothed complexity is upper bounded by $O(σ^{-1/2} d^{11/4} \log(n)^{7/4})$ pivot steps. Furthermore, we prove a matching high-probability lower bound of $Ω( σ^{-1/2} d^{1/2}\ln(4/σ)^{-1/4})$ on the combinatorial diameter of the feasible polyhedron after smoothing, on instances using $n = \lfloor (4/σ)^d \rfloor$ inequality constraints. This lower bound indicates that our algorithm has optimal noise dependence among all simplex methods, up to polylogarithmic factors.

翻译：平滑分析是一种用于分析算法性能的方法，尤其适用于那些在实践中运行时间显著优于最坏情况分析所证明结果的算法。Spielman 和 Teng（STOC '01）提出了算法分析的平滑分析框架，并将其应用于单纯形方法。给定一个具有 $d$ 个变量和 $n$ 个不等式约束的任意线性规划，Spielman 和 Teng 证明单纯形方法的运行时间为 $O(σ^{-30} d^{55} n^{86})$，其中 $σ> 0$ 是添加到原始 LP 数据中的高斯分布噪声的标准差。Spielman 和 Teng 的结果在一系列工作中得到了简化和加强，目前最强的上界是 Huiberts、Lee 和 Zhang（STOC '23）得到的 $O(σ^{-3/2} d^{13/4} \log(n)^{7/4})$ 次枢轴步骤。我们证明存在一种单纯形方法，其平滑复杂度上界为 $O(σ^{-1/2} d^{11/4} \log(n)^{7/4})$ 次枢轴步骤。此外，我们证明了在平滑处理后的可行多面体的组合直径上，对于使用 $n = \lfloor (4/σ)^d \rfloor$ 个不等式约束的实例，存在一个匹配的高概率下界 $Ω( σ^{-1/2} d^{1/2}\ln(4/σ)^{-1/4})$。该下界表明，在所有单纯形方法中，我们的算法具有最优的噪声依赖性（最多相差多对数因子）。