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

Abstract: We consider the optimal transmission of distributed correlated discrete memoryless sources across a network with capacity constraints. We present several previously undiscussed structural properties of the set of feasible rates and transmission schemes. We extend previous results concerning the intersection of polymatroids and contrapolymatroids to characterize when all of the vertices of the Slepian-Wolf rate region are feasible for the capacity constrained network. An explicit relationship between the conditional independence relationships of the distributed sources and the number of vertices for the Slepian-Wolf rate region are given. These properties are then applied to characterize the optimal transmission rate and scheme and its connection to the corner points of the Slepian-Wolf rate region. In particular, we demonstrate that when the per-source compression costs are in tension with the per-link flow costs the optimal flow/rate point need not coincide with a vertex of the Slepian-Wolf rate region. Finally, we connect results for the single-sink problem to the multi-sink problem by extending structural insights and developing upper and lower bounds on the optimal cost of the multi-sink problem.