A quick estimate for the volume of a polyhedron

TL;DR

This paper introduces a simple, efficient approximation formula for the volume of certain polyhedra, providing the best known estimates for transportation polytopes within a factor related to their codimension.

Contribution

It presents a novel, computationally efficient formula for estimating the volume of polyhedra defined as intersections of the non-negative orthant and affine subspaces, with proven approximation bounds.

Findings

The formula approximates the volume within a factor of rac{1}{ ext{constant}}^m.

It provides the best known estimates for transportation polytopes.

The approximation is computationally efficient.

Abstract

Let $P$ be a bounded polyhedron defined as the intersection of the non-negative orthant ${\Bbb R}^n_+$ and an affine subspace of codimension $m$ in ${\Bbb R}^n$. We show that a simple and computationally efficient formula approximates the volume of $P$ within a factor of $\gamma^m$, where $\gamma >0$ is an absolute constant. The formula provides the best known estimate for the volume of transportation polytopes from a wide family.