A Deterministic Algorithm of Quasi-Polynomial Complexity for Clipped Cubes Volume Approximation

TL;DR

This paper presents a deterministic quasi-polynomial algorithm for approximating the volume of hypercube intersections with certain sets, including non-convex shapes like balls and ellipsoids, with detailed analysis for specific cases.

Contribution

It introduces a novel quasi-polynomial complexity method for volume approximation of hypercube intersections with polynomial-defined sets, extending to non-convex shapes.

Findings

Algorithm achieves quasi-polynomial complexity

Method applies to non-convex sets like balls and ellipsoids

Provides convergence and complexity analysis for specific cases

Abstract

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 compute the volume of the hypercube intersected with a fixed number of sets, described by equations of the form $\sum_{q=1}^n a_q(x_q) \leq b$, where $a_q : \mathbb{R} \to \mathbb{R}$ are polynomial functions and $b \in \mathbb{R}$. Note that the resulting sets are not necessarily convex. This type of equations describe, among others, half-spaces, balls and ellipsoids. We give detailed convergence and complexity analysis for the case in which the unit hypercube is clipped by balls of arbitrary radius but with centers whom distance to the unit hypercube is greater than $1$ (one).