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

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).