New Bounds for the Vertices of the Integer Hull

TL;DR

This paper establishes new bounds on the support size of vertices of the integer hull in linear programming, using probabilistic techniques, and provides algorithms for enumeration with nearly optimal bounds.

Contribution

It introduces a probabilistic approach to bounding the support size of integer hull vertices, improving existing bounds and offering enumeration algorithms.

Findings

New bounds on the support size of vertices matching or improving previous bounds.

A generalized Hoeffding bound for vector-valued random variables.

An enumeration algorithm for vertices with near-optimal bounds.

Abstract

The vertices of the integer hull are the integral equivalent to the well-studied basic feasible solutions of linear programs. In this paper we give new bounds on the number of non-zero components -- their support -- of these vertices matching either the best known bounds or improving upon them. While the best known bounds make use of deep techniques, we only use basic results from probability theory to make use of the concentration of measure effect. To show the versatility of our techniques, we use our results to give the best known bounds on the number of such vertices and an algorithm to enumerate them. We also improve upon the known lower bounds to show that our results are nearly optimal. One of the main ingredients of our work is a generalization of the famous Hoeffding bound to vector-valued random variables that might be of general interest.