### Extension Complexity Part II - Classifying Nonnegative Rank by Randomized Communication

A classical result from Yannakakis states that $$\text{xc}(P) = \text{rank}_+(S),$$ where \(P\) is a polytope with slack matrix \(S\). Yannakakis further showed that with nondeterministic communication complexity, one can give a lower bound for the nonnegative rank. This lower bound is sufficient to prove that some polytopes, like the correlation polytope, have exponential extension complexity. However, it fails for many other examples. In a 2013 paper, "Extended formulations, nonnegative factorizations, and randomized communication protocols," by Faenza, Fiorini, Grappe, and Tiwary, they show that a model of randomized communication complexity can not only lower bound nonnegative rank, but exactly characterize it. In my communication complexity reading group, I gave a talk presenting this paper and its results. The proofs are extremely elegant and simple, making for a very enjoyable read. For those interested, here is a link to my lecture notes!