The Mixing Rate of Markov Chains, an Isoperimetric Inequality, and Computing the Volume

Sinclair and Jerrum derived a bound on the mixing rate of time-reversible Markov chains in terms of their conductance. We generalize this result, not assuming time-reversibility and using a weaker notion of conductance. We prove an isoperimetric inequality for subsets of a convex body. These results are combined to simplify an algorithm of Dyer, Frieze and Kannan for approximating the volume of a convex body, and to improve running time bounds. 0. Introduction and preliminaries Recently Dyer, Frieze and Kannan (1989) designed a polynomial time randomized algorithm to a p proximate the volume of a convex body K in R”. A crucial step of the algorithm is to generate a random point in a convex body. This is achieved by making a random walk on the lattice points inside the body. The analysis of the algorithm depends on two factors: a theorem of Sinclair and Jerrum (1988) on the mixing rate of time-reversible Markov chains and on an isoperimetric inequality for subsets of a convex body. In this paper we improve both of these steps. We generalize the theorem of Sinclair and Jerrum (1988). In fact, we introduce a new proof technique to handle the mixing rate of a Markov chain. This will allow us to handle Markov chains in which the “small” sets need not have large conductance. As a byproduct, we can drop the time-reversibility assumption (this will not help us in the current application, however), and obtain a sharper bound for the mixing rate depending on the starting distribution. Dyer, Fkieze and Kannan point out that an improvement in their isoperimetric inequality, in particular the removal of the step of approximating the body by one with bounded curvature, would result in simpler and faster algorithms. We give a fairly simple proof of such an improved isoperimetric inequality. Mikl6s Simonovits Mathematical Research Institute Hungarian Academy of Science, Budapest and Rutgers University, New Brunswick, N J Then we sketch how these results can be a p plied to modify the algorithm of Dyer, Frieze and Kannan. In particular, we improve the running time: to approximate the volume of a convex body in R” with relative error less than E and probability of error less than 6, their algorithm has to solve O(n23(log n ) ‘ ~ ~ log( I/&) log( 1/6)) convex programs; our one needs only O(n16(log n)6 log(n/&) l o g ( n / 6 ) ~ ~ ) simple membership tests. While this is still far from being practical, there is hope of further improvements. Acknowledgements. We are glad to acknowledge discussions on the topic of this paper with Ravi Kannan, Mike Steele, and Doma Szaisz. We are particularly greatful to Imre B k h y for pointing out an error in an earlier version of the manuscript. Preliminaries. A convez body is a compact and fulldimensional convex set in R”. For two convex bodies K1 and K2, we consider their Minkowski sum: and also the (less standard) notation K1K2 = {Z E R” : K2 + z G Ki}. For algorithmic purposes, a standard way to describe a convex body K as an input is a well-guaranteed weak separation oracle, which means the following: Definition (Weak separation oracle). For any y E Qfi we may ask the oracle whether y belongs to K or not; together with this query, we also include an error tolerance 6 > 0. The answer will be “YES” or “NO”. The “YES” answer means that the distance of y from K is at most 6; the “NO” answer means that the distance of y from R” \ K is less than 6. In this case, we also require a “proof” of this fact, in the form a 346 CH2925-6/90/0000/0346$01 .OO