Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Black, Alexander E., Criado, Francisco
Institutions: University of California, Davis, CUNEF Universidad
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4399598121