  We consider the following communication task in the multi-party setting,
which involves a joint random variable $XYZMN$ with the property that $M$ is
independent of $YZN$ conditioned on $X$ and $N$ is independent of $XZM$
conditioned on $Y$. Three parties Alice, Bob and Charlie, respectively, observe
samples $x,y$ and $z$ from $XYZ$. Alice and Bob communicate messages to Charlie
with the goal that Charlie can output a sample from $MN$ having correct
correlation with $XYZ$. This task reflects the simultaneous message passing
model of communication complexity. Furthermore, it is a generalization of some
well studied problems in information theory, such as distributed source coding,
source coding with a helper and one sender and one receiver message
compression. It is also closely related to the lossy distributed source coding
task.
  Our main result is an achievable communication region for this task in the
one-shot setting, through which we obtain a near optimal characterization using
auxiliary random variables of bounded size. We employ our achievability result
to provide a near-optimal one-shot communication region for the task of lossy
distributed source coding, in terms of auxiliary random variables of bounded
size. Finally, we show that interaction is necessary to achieve the optimal
expected communication cost for our main task.
