Thin Partitions: Isoperimetric Inequalities and Sampling Algorithms for some Nonconvex Families

TL;DR

This paper introduces an efficient polynomial-time sampling algorithm for star-shaped bodies, a significant nonconvex generalization of convex sets, based on new isoperimetric inequalities and applicable to volume computation.

Contribution

The paper presents the first polynomial-time sampling and volume algorithms for star-shaped bodies using novel isoperimetric inequalities for nonconvex domains.

Findings

Polynomial-time sampling algorithm for star-shaped bodies

New isoperimetric inequality for nonconvex domains

Polynomial volume computation method for star-shaped sets

Abstract

Star-shaped bodies are an important nonconvex generalization of convex bodies (e.g., linear programming with violations). Here we present an efficient algorithm for sampling a given star-shaped body. The complexity of the algorithm grows polynomially in the dimension and inverse polynomially in the fraction of the volume taken up by the kernel of the star-shaped body. The analysis is based on a new isoperimetric inequality. Our main technical contribution is a tool for proving such inequalities when the domain is not convex. As a consequence, we obtain a polynomial algorithm for computing the volume of such a set as well. In contrast, linear optimization over star-shaped sets is NP-hard.