Beyond Smoothed Analysis: Analyzing the Simplex Method by the Book

TL;DR

This paper introduces a new 'by the book' analysis framework that models both input data and algorithms to better explain practical performance, and applies it to prove polynomial time for the simplex method under realistic assumptions.

Contribution

The paper proposes a novel analysis framework that incorporates algorithm design principles and input modeling, providing a more accurate theoretical understanding of practical algorithm behavior.

Findings

The framework models both input data and algorithm behavior.

Under realistic assumptions, the simplex method runs in polynomial time.

The approach overcomes limitations of smoothed analysis.

Abstract

Narrowing the gap between theory and practice is a longstanding goal of the algorithm analysis community. To further progress our understanding of how algorithms work in practice, we propose a new algorithm analysis framework that we call by the book analysis. In contrast to earlier frameworks, by the book analysis not only models an algorithm's input data, but also the algorithm itself. Results from by the book analysis are meant to correspond well with established knowledge of an algorithm's practical behavior, as they are meant to be grounded in observations from implementations, input modeling best practices, and measurements on practical benchmark instances. We apply our framework to the simplex method, an algorithm which is beloved for its excellent performance in practice and notorious for its high running time under worst-case analysis. The simplex method similarly showcased the state of the art framework smoothed analysis (Spielman and Teng, STOC'01). We explain how our framework overcomes several weaknesses of smoothed analysis and we prove that under input scaling assumptions, feasibility tolerances and other design principles used by simplex method implementations, the simplex method indeed attains a polynomial running time.