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The capacity or approximations to capacity of various single-sourcesingle-destination relay network models has been characterized in terms of thecut-set upper bound. In principle, a direct computation of this bound requiresevaluating the cut capacity over exponentially many cuts. We show that theminimum cut capacity of a relay network under some special assumptions can becast as a minimization of a submodular function, and as a result, can becomputed efficiently. We use this result to show that the capacity, or anapproximation to the capacity within a constant gap for the Gaussian, wirelesserasure, and Avestimehr-Diggavi-Tse deterministic relay network models can becomputed in polynomial time. We present some empirical results showing thatcomputing constant-gap approximations to the capacity of Gaussian relaynetworks with around 300 nodes can be done in order of minutes.  For Gaussian networks, cut-set capacities are also functions of the powersassigned to the nodes. We consider a family of power optimization problems andshow that they can be solved in polynomial time. In particular, we show thatthe minimization of the sum of powers assigned to the nodes subject to aminimum rate constraint (measured in terms of cut-set bounds) can be computedin polynomial time. We propose an heuristic algorithm to solve this problem andmeasure its performance through simulations on random Gaussian networks. Weobserve that in the optimal allocations most of the power is assigned to asmall subset of relays, which suggests that network simplification may bepossible without excessive performance degradation.

Efficient Capacity Computation and Power Optimization for Relay Networks