An LP lower bound for rate distortion with variable side information

We consider a rate distortion problem with side information at multiple decoders. Several lower bounds have been proposed for this general problem or special cases of it. We provide a lower bound for general instances of this problem, which was inspired by a linear-programming lower bound for index coding, and show that it subsumes most of the lower bounds in literature. Using this bound, we explicitly characterize the rate distortion function of a problem which can be seen as a Gaussian analogue of the “odd-cycle” index coding problem.