Polynomial-time algorithm for approximating the volume of convex bodies

The paper is a joint work by Martin Dyer, Alan M. Frieze and Ravindran Kannan.^[1]

The main result of the paper is a randomized algorithm for finding an $\epsilon$ approximation to the volume of a convex body $K$ in $n$-dimensional Euclidean space by assuming the existence of a membership oracle. The algorithm takes time bounded by a polynomial in $n$, the dimension of $K$ and $1/\epsilon$.

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 $K$ by placing a grid consisting $n$-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.

References

1. 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.