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Beyond Smoothed Analysis: Analyzing the Simplex Method by the Book

Eleon Bach, Alexander E. Black, Sophie Huiberts, Sean Kafer

TL;DR

This work introduces by-the-book analysis, a framework that grounds algorithmic analysis in practical implementations and data, and applies it to the simplex method. By modeling bound perturbations, scaling, and tolerance choices observed in real solvers, the authors derive a polynomial running-time bound for a two-phase simplex method under these realistic assumptions, leveraging the shadow vertex path and mean width of the feasible set. The key contributions include a detailed running-time analysis that remains feasible for sparse LPs, a Phase I/Phase II procedure incorporating bound perturbations, and a corollary bounding pivot steps through a combination of slack and multiplier properties along semi-random shadow paths. The results bridge theory and practice by tying theoretical guarantees to measurable, well-scoped practical parameters, highlighting the importance of scaling and perturbation strategies in industrial LP solvers.

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.

Beyond Smoothed Analysis: Analyzing the Simplex Method by the Book

TL;DR

This work introduces by-the-book analysis, a framework that grounds algorithmic analysis in practical implementations and data, and applies it to the simplex method. By modeling bound perturbations, scaling, and tolerance choices observed in real solvers, the authors derive a polynomial running-time bound for a two-phase simplex method under these realistic assumptions, leveraging the shadow vertex path and mean width of the feasible set. The key contributions include a detailed running-time analysis that remains feasible for sparse LPs, a Phase I/Phase II procedure incorporating bound perturbations, and a corollary bounding pivot steps through a combination of slack and multiplier properties along semi-random shadow paths. The results bridge theory and practice by tying theoretical guarantees to measurable, well-scoped practical parameters, highlighting the importance of scaling and perturbation strategies in industrial LP solvers.

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.
Paper Structure (30 sections, 18 theorems, 124 equations, 1 figure)

This paper contains 30 sections, 18 theorems, 124 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Smoothed Analysis and the Simplex Method
  3. Limitations of Smoothed Analysis
  4. Introducing By The Book Analysis
  5. Observations of Practical LP
  6. Scaling
  7. Tolerances
  8. Perturbations
  9. Overview
  10. Two-Phase Simplex Method
  11. Running time bound
  12. Proof overview
  13. Related Work
  14. Observations
  15. Glop
  16. ...and 15 more sections

Key Result

Lemma 2

Let $\eta, \gamma \in \mathbb{R}_{>0}$, and let ${\hbox{\boldmath$\bf v$}} \in \mathbb R^k$. Let $\hat{{\hbox{\boldmath$\bf v$}}}$ be a $({\hbox{\boldmath$\bf v$}}, \eta,\gamma)$-exponentially distributed random vector. Then

Figures (1)

  • Figure 1: Schematic depiction of by the book analysis

Theorems & Definitions (44)

  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 34 more