Two lossy source coding problems with causal side-information

Single-letter characterisations of the admissible rate regions of the Gu-Effros two-hop network and the Gray-Wyner network with side-information are open problems. We show that both problems permit single-letter solutions under the assumption of causal side-information. In particular, the special structure of the causal side-information decoder allows one to match converse theorems to known coding theorems using standard information-theoretic tools. This observation complements similar results by Maor and Merhav for the Heegard-Berger problem and the successive-refinement problem with side-information, and suggests that more general results for causal side-information networks may be possible.