Diamond Message Set Groupcasting: From an Inner Bound for the DM Broadcast Channel to the Capacity Region of the Combination Network

Multiple groupcasting over the broadcast channel (BC) is studied in a special setting. In particular, an inner bound is obtained for the
$K$
-receiver discrete memoryless (DM) BC for the diamond message set which consists of four groupcast messages: one desired by all receivers, one by all but two receivers, and two more desired by all but each one of those two receivers. The inner bound is based on rate-splitting and superposition coding and is given in explicit form herein as a union over coding distributions of four-dimensional polytopes. This inner bound is then shown to be the capacity for a certain class of partially ordered DM BCs with order defined via the less noisy condition. When specialized to the so- called combination network, which is a class of three-layer (two-hop) broadcast networks parameterized by
$2^{K}{-}1$
finite-and-arbitrary-capacity noiseless links from the source node in the first layer to as many nodes of the second layer, our top-down approach from the DM BC to the combination network yields an explicit inner bound as a single polytope via the identification of a single coding distribution. This inner bound consists of inequalities which are then identified to be within the class of generalized cut-set outer bounds recently obtained by Salimi et al for broadcast networks. We hence establish the capacity region of the general
$K$
-user combination network for the diamond message set, and do so in explicit and structured form. Such a result implies a certain strength of our inner bound for the DM BC in that it (a) produces a hitherto unknown capacity region when specialized to the combination network and (b) may capture many combinatorial aspects of the capacity region of the
$K$
-receiver DM BC itself for the diamond message set. Moreover, we extend that inner bound by adding binning to it. As in the no-binning case, we provide the more general inner bound in explicit form.