  In this paper, distributed (or multiterminal) source coding with one
distortion criterion and correlated messages is considered. This problem can be
also called ``Berger-Yeung problem with correlated messages''. It corresponds
to the source coding part of the graph-based framework for transmission of a
pair of correlated sources over the multiple-access channel (MAC) where one is
lossless and the other is lossy. As a result, the achievable rate-distortion
region for this problem is provided. It is an information-theoretic
characterization of the rate of exponential growth (as a function of the number
of source samples) of the size of the bipartite graphs which can represent a
pair of correlated sources with satisfying one distortion criterion. A rigorous
proof of the achievability and the converse part is given. It is also shown
that there exists functional duality between Berger-Yeung problem with
correlated messages and semi-deterministic broadcast channel with correlated
messages. This means that the optimal encoder-decoder mappings for one problem
become the optimal decoder-encoder mappings for the dual problem. In the
duality setup, the correlation structure of the messages in the two dual
problems, source distortion measure and channel cost measure are also
specified.
