Optimal Smoothed Analysis of the Simplex Method
Eleon Bach, Sophie Huiberts
TL;DR
This paper advances the smoothed analysis of the simplex method by proving a near-optimal bound of $O(σ^{-1/2} d^{11/4} \log(n)^{7/4})$ pivot steps under Gaussian perturbations, improving upon all prior bounds in the noise parameter $σ$ while maintaining polylog dependence on the dimension and constraints. A key innovation is a three-phase shadow-vertex algorithm that employs a semi-random shadow plane, enabling a direct primal-space shadow analysis and reducing dependence on the perturbation magnitude. The authors introduce the semi-random shadow size $R(n,d,σ)$ and show $R(n,d,σ) \leq O\left(\sqrt{σ^{-1} \sqrt{d^{11} \log(n)^7}} + d^3 \log(n)^2 \right)$, along with a matching high-probability lower bound on the combinatorial diameter under smoothing, which demonstrates optimal noise dependence up to polylog factors. Together, these results sharpen the theoretical understanding of why simplex-type methods perform well in practice and provide tighter benchmarks for smoothed analyses of LP algorithms. The techniques combine random objective perturbations with algorithmically injected randomness (semi-random shadows), auxiliary LP reductions, and intricate geometric-probabilistic arguments to control shadow-path lengths and vertex separation.
Abstract
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.