Random Shadows of Fixed Polytopes

TL;DR

This paper establishes new upper bounds on the expected number of vertices in random shadows of fixed polytopes, linking geometric parameters to shadow complexity, with tight bounds demonstrated for zonotopes.

Contribution

It provides the first comprehensive bounds on shadow vertex counts based on various polytope parameters, advancing understanding of shadow complexity in linear programming.

Findings

Upper bounds on expected shadow vertices in terms of diameter and edge lengths

Lower bounds matching upper bounds for zonotopes

Tight bounds established for specific polytope classes

Abstract

Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove multiple upper bounds on the expected number of vertices of a random shadow of a fixed polytope. Our bounds are in terms of various parameters in the literature including geometric diameter and edge lengths, minimal and maximal slack, maximal coordinates for lattice polytopes, and maximum absolute values of subdeterminants. For the case of geometric diameter and edge lengths, we prove lower bounds and argue that our upper and lower bounds are both tight for zonotopes.