Related papers: Computing the Volume of Polytropes
We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We…
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…
We survey the computation of polytope volumes by the algorithms of Normaliz to which the Lawrence algorithm has recently been added. It has enabled us to master volume computations for polytopes from social choice in dimension $119$. This…
Volume computation for $d$-polytopes $\mathcal{P}$ is fundamental in mathematics. There are known volume computation algorithms, mostly based on triangulation or signed-decomposition of $\mathcal{P}$. We consider $…
Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex polytopes exactly equals the normalized…
The volume is an important attribute of a convex body. In general, it is quite difficult to calculate the exact volume. But in many cases, it suffices to have an approximate value. Volume estimation methods for convex bodies have been…
This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of…
We consider a minimum enclosing and maximum inscribed tropical balls for any given tropical polytope over the tropical projective torus in terms of the tropical metric with the max-plus algebra. We show that we can obtain such tropical…
In this note, a new method for deriving the volume of hypersphere is proposed by using probability theory. The explicit expression of the multiple times convolution of the probability density functions we should use is very complicated. But…
We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex…
Answering a question of Gamarnik and Smedira, we give a polynomial time algorithm that approximately computes the volume of a truncation of a relaxation of the independent set polytope, improving on their quasi-polynomial time algorithm.…
We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a…
The aim of this work is to introduce several different volume computation methods of the graph polytope associated with various type of finite simple graphs. Among them, we obtained the recursive volume formula (RVF) that is fundamental and…
The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which…
We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the…
There is an elegant expression for the volume of hypercube $[0,1]^n$ clipped by a single hyperplane. In the article the formula is generalized to the case of more than one hyperplane. An important foundation for the result is Lawrence's…
Computing the volume of a polytope in high dimensions is computationally challenging but has wide applications. Current state-of-the-art algorithms to compute such volumes rely on efficient sampling of a Gaussian distribution restricted to…
Lattice polytope representation of natural numbers is introduced based on the fundamental theorem of arithmetic. The combinatorial and geometric properties of the polytopes are studied using Polymake and Qhull software. The volume of the…
We give a deterministic method of quasi-polynomial complexity to approximate the volume of the intersection of the unit hypercube with two specific sets. The method can actually be applied (without losing the quasi-polynomial complexity) to…
Union volume estimation is a classical algorithmic problem. Given a family of objects $O_1,\ldots,O_n \subseteq \mathbb{R}^d$, we want to approximate the volume of their union. In the special case where all objects are boxes (also known as…