Thin Partitions: Isoperimetric Inequalities and Sampling Algorithms for some Nonconvex Families
Data Structures and Algorithms
2009-04-06 v1 Functional Analysis
Probability
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.
Cite
@article{arxiv.0904.0583,
title = {Thin Partitions: Isoperimetric Inequalities and Sampling Algorithms for some Nonconvex Families},
author = {Karthekeyan Chandrasekaran and Daniel Dadush and Santosh Vempala},
journal= {arXiv preprint arXiv:0904.0583},
year = {2009}
}