Polynomial-time algorithm for approximating the volume of convex bodies

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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 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.

References

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