  We study a network utility maximization (NUM) decomposition in which the set
of flow rates is grouped by source-destination pairs. We develop theorems for
both single-path and multipath cases, which relate an arbitrary NUM problem
involving all flow rates to a simpler problem involving only the aggregate
rates for each source-destination pair. The optimal aggregate flows are then
apportioned among the constituent flows of each pair. This apportionment is
simple for the case of $\alpha$-fair utility functions. We also show how the
decomposition can be implemented with the alternating direction method of
multipliers (ADMM) algorithm.
