The Niceness of Unique Sink Orientations

TL;DR

This paper systematically studies the niceness of Unique Sink Orientations (USO) to understand the behavior of the Random Edge pivot rule in the simplex algorithm, providing bounds, derandomization, and algorithmic insights.

Contribution

It settles open questions about the niceness of acyclic USO, establishes tight bounds for Random Edge, and introduces a derandomization approach with similar bounds.

Findings

Niceness implies natural upper bounds for Random Edge.

Random Edge is polynomial on many USO, including cyclic ones.

A derandomized version of Random Edge achieves similar bounds.

Abstract

Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of \emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least $n^{\Omega(2^n)}$ many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness and discuss some algorithmic properties of the reachmap.