Structural and Optimization Properties for Joint Selection of Source Rates and Network Flow

We consider the optimal transmission of distributedcorrelated discrete memoryless sources across a network withcapacity constraints. We present several previously undiscussedstructural properties of the set of feasible rates and transmissionschemes. We extend previous results concerning the intersectionof polymatroids and contrapolymatroids to characterize whenall of the vertices of the Slepian-Wolf rate region are feasiblefor the capacity constrained network. An explicit relationshipbetween the conditional independence relationships of the dis-tributed sources and the number of vertices for the Slepian-Wolf rate region are given. These properties are then applied tocharacterize the optimal transmission rate and scheme and itsconnection to the corner points of the Slepian-Wolf rate region. Inparticular, we demonstrate that when the per-source compressioncosts are in tension with the per-link flow costs the optimalflow/rate point need not coincide with a vertex of the Slepian-Wolfrate region. Finally, we connect results for the single-sink problemto the multi-sink problem by extending structural insights anddeveloping upper and lower bounds on the optimal cost of themulti-sink problem.