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

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.

💡 Research Summary

The paper investigates algorithmic and geometric properties of star‑shaped sets, a natural non‑convex generalization of convex bodies that arise, for example, in linear programming with violations. A star‑shaped set (S\subset\mathbb{R}^d) possesses a non‑empty kernel (\mathcal K(S)) – the set of points from which every line segment to any point of (S) stays inside (S). The authors denote by (\eta = \operatorname{vol}(\mathcal K(S))/\operatorname{vol}(S)) the fraction of total volume occupied by the kernel. Their main contributions are (i) a new isoperimetric inequality that holds for arbitrary star‑shaped bodies, (ii) a “thin‑partition” technique that enables the inequality, (iii) a polynomial‑time random‑walk based sampling algorithm whose running time is polynomial in the dimension (d) and inverse‑polynomial in (\eta), and (iv) a proof that linear optimization over star‑shaped sets is NP‑hard, thereby separating the complexity of sampling/volume estimation from that of optimization in this non‑convex regime.

Thin partitions and isoperimetry.
The authors introduce a geometric decomposition of the ambient space into a collection of “thin slabs” (or layers) such that each slab has thickness on the order of (1/\sqrt{d}). Within a slab the set behaves almost like a convex body because any two points can be connected by a line segment that either stays inside the slab or passes through the kernel. By carefully analyzing the surface measure of the boundary between a slab and its complement, they prove the following conductance lower bound for the uniform distribution on (S): \