Optimal Smoothed Analysis of the Simplex Method

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 $O(\sigma^{-30} d^{55} n^{86})$, where $\sigma > 0$ 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 $O(\sigma^{-3/2} d^{13/4} \log(n)^{7/4})$ 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 $O(\sigma^{-1/2} d^{11/4} \log(n)^{7/4})$ pivot steps. Furthermore, we prove a matching high-probability lower bound of $\Omega( \sigma^{-1/2} d^{1/2}\ln(4/\sigma)^{-1/4})$ on the combinatorial diameter of the feasible polyhedron after smoothing, on instances using $n = \lfloor (4/\sigma)^d \rfloor$ inequality constraints. This lower bound indicates that our algorithm has optimal noise dependence among all simplex methods, up to polylogarithmic factors.