A Note on Bounding the Mixing Time by Linear Programming

Abstract

We relate the mixing time of Markov chains on a partial order with unique minimal and maximal elements to the solution of associated linear programs. The linear minimization program we construct has one variable per state and (the square of) its solution is an upper bound on the mixing time. The proof of this theorem uses the coupling technique and a generalization of the distance function commonly used in this context. Explicit solutions are obtained for the simple Markov chains on the hypercube and on the independent sets of a complete bipartite graph.

As an application we define new a Markov chain on the down-sets (ideals) of a partial order for which our technique yields a simple proof of rapid mixing, provided that in the Hasse-graph of the partial order the number of elements at distance at most 2 from any given element is bounded by 4. This chain is a variation of the Luby-Vigoda chain on independent sets, which can also be used directly to sample down-sets, but our result applies to a larger class of partial orders.