Source Coding for Synthesizing Correlated Randomness

We consider a scenario wherein two parties Alice and Bob are provided X-1(n) and X-2(n) - samples that are IID from a PMF p(X1X2). Alice and Bob can communicate to Charles over (noiseless) communication links of rate R-1 and R-2 respectively. Their goal is to enable Charles generate samples Y-n such that the triple (X-1(n), X-2(n), Y-n) has a PMF that is close, in total variation, to Pi p(X1X2Y). In addition, the three parties may posses shared common randomness at rate C. We address the problem of characterizing the set of rate triples (R-1, R-2, C) for which the above goal can be accomplished. We provide a set of sufficient conditions, i.e., an achievable rate region for this three party setup. Our work also provides a complete characterization of a point-to-point setup wherein Bob is absent and Charles is provided with side-information.