Random walks on the vertices of transportation polytopes with constant number of sources

We consider the problem of uniformly sampling a vertex of a transportation polytope with m sources and n destinations, where m is a constant. We analyse a natural random walk on the edge-vertex graph of the polytope. The analysis makes use of the multi-commodity flow technique of Sinclair [20] together with ideas developed by Morris and Sinclair [15, 16] for the knapsack problem, and Cryan et al. [2] for contingency tables, to establish that the random walk approaches the uniform distribution in time n(O(m2)).