A new algorithm for the volume of a convex polytope
J. B. Lasserre, E. S. Zeron
TL;DR
This paper addresses the exact computation of the volume of a convex polytope defined by half-space constraints $\Omega(y)=\{x\in\mathbb{R}^n_+\,|\,Ax\le y\}$ by treating the volume as a function $g(y)$ and deriving its Laplace transform $G(\lambda)$. It proves $G(\lambda)=\frac{1}{\prod_{i=1}^m \lambda_i}\frac{1}{\prod_{j=1}^n (A'\lambda)_j}$ for $\Re(\lambda)>0$ and $\Re(A'\lambda)>0$, with $g(y)$ recovered via the $m$-fold inverse Laplace transform using multi-dimensional Cauchy residues. The authors present two algorithms—the direct method and the associated-transform method—that perform the inverse transform by iterative one-dimensional residue calculations; both achieve $O(n^m)$ complexity, making them attractive when $n$ is large and $m$ is small. This approach offers an exact, dual perspective to existing $O(m^n)$ methods that operate in the primal space, and it provides practical strategies for pole placement and path selection to ensure robust computation.
Abstract
We provide two algorithms for computing the volume of a convex polytope with half-space representation {x>=0; Ax <=b} for some (m,n) matrix A and some m-vector b. Both algorithms have a O(n^m) computational complexity which makes them especially attractive for large n and relatively small m when the other methods with O(m^n) complexity fail. The methodology which differs from previous existing methods uses a Laplace transform technique that is well-suited to the half-space representation of the polytope.