New Bounds for the Vertices of the Integer Hull

by   Sebastian Berndt, et al.

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.



There are no comments yet.


page 1

page 2

page 3

page 4


Sharper bounds for uniformly stable algorithms

The generalization bounds for stable algorithms is a classical question ...

Lower bounds for prams over Z

This paper presents a new abstract method for proving lower bounds in co...

Computing Permanents on a Trellis

The problem of computing the permanent of a matrix has attracted interes...

An almost optimal bound on the number of intersections of two simple polygons

What is the maximum number of intersections of the boundaries of a simpl...

Counting vertices of integer polytopes defined by facets

We present a number of complexity results concerning the problem of coun...

On the Complexity of the k-Level in Arrangements of Pseudoplanes

A classical open problem in combinatorial geometry is to obtain tight as...

Expected capture time and throttling number for cop versus gambler

We bound expected capture time and throttling number for the cop versus ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.