A friendly smoothed analysis of the simplex method

Abstract

Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng (JACM ‘04), who the developed the notion of smoothed analysis. Starting from an arbitrary linear program with d variables and n constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP (smoothed LP), where variance σ Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected O(n86 d55 σ−30) number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by SpielmanDeshpande (FOCS ‘05) and later Vershynin (SICOMP ‘09). The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected O(d3 log3 n σ−4 + d9log7 n) number of pivots, improving the dependence on n from polynomial to logarithmic.