Thin Partitions: Isoperimetric Inequalities and a Sampling Algorithm for Star Shaped Bodies

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.

• Type: Journal
• Year: 2010
• Where: SODA
• Authors: Karthekeyan Chandrasekaran, Daniel Dadush, Santosh Vempala
• Keywords: Algorithm, Algorithms, SODA 2010, Star-shaped, Star-shaped Body