Polynomial-time algorithm for approximating the volume of convex bodies
The main result of the paper is a randomized algorithm for finding an approximation to the volume of a convex body in -dimensional Euclidean space by assuming the existence of a membership oracle. The algorithm takes time bounded by a polynomial in , the dimension of and .
The algorithm is a sophisticated usage of the so-called Markov chain Monte Carlo (MCMC) method. The basic scheme of the algorithm is a nearly uniform sampling from within by placing a grid consisting -dimensional cubes and doing a random walk over these cubes. By using the theory of rapidly mixing Markov chains, they show that it takes a polynomial time for the random walk to settle down to being a nearly uniform distribution.
- M.Dyer, A.Frieze and R.Kannan (1991). "A random polynomial-time algorithm for approximating the volume of convex bodies". Journal of the ACM. 38 (1): 1–17. doi:10.1145/102782.102783.