Public-private separation in linear network-coded simultaneous multicast and unicast

We consider a network coding problem with a single source and multiple sinks. A common set of messages is demanded by all sinks. In addition, each sink demands a private set of messages, which is disjoint from the private set of all other sinks. This pattern of demands is called simultaneous multicast and unicast. This is a specific case of the general connections problem, for which determining the existence of a linear network coding solution is NP-hard. However, the multicast or disjoint broadcast problems, which are the individual components of the simultaneous multicast and unicast problem, are both linearly solvable in polynomial time. Observing that the mincut conditions are insufficient to show the existence of linear network coding solution for simultaneous multicast and unicast, we study a new set of sufficient conditions. We show that public-private separation is the sufficient condition for the existence of linear network coding solution. We start with a set of graphs called 3-level graphs and provide certain extensions.