Flow Decomposition

The decode-forward achievable region is studied for general networks. The region is subject to a fundamental tension in which nodes individually benefit at the expense of others. The complexity of the region depends on all the ways of resolving this tension. Two sets of constraints define an outer-bound on the decode-forward region: first, the conventional mutual-information inequalities implied by the one-relay channel, and second, causality constraints that ensure nodes only forward messages they have already decoded. The framework of flow decomposition is introduced to show these constraints are also sufficient. Flow decomposition provides a way of manipulating regular decode-forward schemes without the long encoding delays and restrictions on bidirectional communication of backward decoding. The two structures that define a flow decomposition are flows and layerings. Flows specify the nodes which encode messages from each source (i.e., the routes) and the encoding delays. Layerings specify the messages decoded at a specific node in the channel. We focus on two types of flow: hierarchical flow, with tree-like routes, and all-cast flow, where each route covers all nodes. For arbitrary flows of either type and any rate-vector satisfying the mutual-information constraints at a specific node, we prove there are equivalent flows and a layering that satisfy both the mutual-information and causality constraints. In separate work, we show that only the mutual-information constraints are active in channels with hierarchical flow, which implies the achievable region has minimal complexity. In channels with all-cast flow, the achievable region is computable.