Zero-Error Function Computation through a Bidirectional Relay

We consider zero error function computation in a three node wireless network. Nodes A and B observe $X$ and $Y$ respectively, and want to compute a function $f(X,Y)$ with zero error. To achieve this, nodes A and B send messages to a relay node C at rates $R_A$ and $R_B$ respectively. The relay C then broadcasts a message to A and B at rate $R_C$ to help them compute $f(X,Y)$ with zero error. We allow block coding, and study the region of rate-triples $(R_A,R_B,R_C)$ that are feasible. The rate region is characterized in terms of graph coloring of some suitably defined probabilistic graphs. We give single letter inner and outer bounds which meet for some simple examples. We provide a sufficient condition on the joint distribution $p_{XY}$ under which the relay can also compute $f(X,Y)$ if A and B can compute it with zero error.