The paper is a joint work by Martin Dyer, Alan M. Frieze and Ravindran Kannan.
The main result of the paper is a randomized algorithm for finding an
ϵ
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
/
ϵ
.
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.
