Network coding for two-unicast with rate (1,2)

We consider a directed acyclic network with two source-sink pairs {s₁, t₁} and {s₂, t₂}. The source s₁ wishes to communicate a message X₁ to the sink t₁ and the source s₂ wishes to communicate two messages X₂ and X₃ to the sink t₂, where X_i, i = 1,2,3, are independent random variables of unit rate. We give a simple characterization for linear solvability of such networks under the condition that the minimum cut from {s₁, s₂} to t₂ equals 3. We develop a region decomposition method for proving this result, which we believe can be an effective approach for non-multicast network coding problem.