Computational geometry: an introduction
Computational geometry: an introduction
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
Sampling and integration of near log-concave functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
On geometric optimization with few violated constraints
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
k-Violation linear programming
Information Processing Letters
A new algorithm for minimizing convex functions over convex sets
Mathematical Programming: Series A and B
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
Solving convex programs by random walks
Journal of the ACM (JACM)
Low-Dimensional Linear Programming with Violations
SIAM Journal on Computing
SIAM Journal on Computing
Simulated annealing in convex bodies and an O*(n4) volume algorithm
Journal of Computer and System Sciences - Special issue on FOCS 2003
The geometry of logconcave functions and sampling algorithms
Random Structures & Algorithms
An Integer Programming Approach for Linear Programs with Probabilistic Constraints
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Approximating the volume of unions and intersections of high-dimensional geometric objects
Computational Geometry: Theory and Applications
On sampling from multivariate distributions
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
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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 polynomi-ally 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.