Smoothed Analysis of Algorithms: Why the Simplex Algorithm Usually Takes Polynomial Time

TL;DR

This paper introduces smoothed analysis, a hybrid approach to evaluate algorithms by considering worst-case inputs subjected to small random perturbations, demonstrating that the simplex algorithm typically runs in polynomial time under this model.

Contribution

It pioneers the smoothed analysis framework and proves that the simplex algorithm has polynomial smoothed complexity, explaining its practical efficiency.

Findings

Simplex algorithm has polynomial smoothed complexity.

Smoothed analysis bridges worst-case and average-case scenarios.

Provides theoretical justification for the practical efficiency of the simplex algorithm.

Abstract

We introduce the smoothed analysis of algorithms, which is a hybrid of the worst-case and average-case analysis of algorithms. In smoothed analysis, we measure the maximum over inputs of the expected performance of an algorithm under small random perturbations of that input. We measure this performance in terms of both the input size and the magnitude of the perturbations. We show that the simplex algorithm has polynomial smoothed complexity.