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The problem of multicasting two nested messages is studied over a class ofnetworks known as combination networks. A source multicasts two messages, acommon and a private message, to several receivers. A subset of the receivers(called the public receivers) only demand the common message and the rest ofthe receivers (called the private receivers) demand both the common and theprivate message. Three encoding schemes are discussed that employ linearsuperposition coding and their optimality is proved in special cases. Thestandard linear superposition scheme is shown to be optimal for networks withtwo public receivers and any number of private receivers. When the number ofpublic receivers increases, this scheme stops being optimal. Two improvementsare discussed: one using pre-encoding at the source, and one using a blockMarkov encoding scheme. The rate-regions that are achieved by the two schemesare characterized in terms of feasibility problems. Both inner-bounds are shownto be the capacity region for networks with three (or fewer) public and anynumber of private receivers. Although the inner bounds are not comparable ingeneral, it is shown through an example that the region achieved by the blockMarkov encoding scheme may strictly include the region achieved by thepre-encoding/linear superposition scheme. Optimality results are founded on thegeneral framework of Balister and Bollob\'as (2012) for sub-modularity of theentropy function. An equivalent graphical representation is introduced and alemma is proved that might be of independent interest.  Motivated by the connections between combination networks and broadcastchannels, a new block Markov encoding scheme is proposed for broadcast channelswith two nested messages. The rate-region that is obtained includes thepreviously known rate-regions. It remains open whether this inclusion isstrict.

Capacity Results for Multicasting Nested Message Sets over Combination Networks