We consider a source that would like to communicate with a destination over a layered Gaussian relay network. We present a computationally efficient method that enables to select a near-optimal (in terms of throughput) subnetwork of a given size connecting the source with the destination. Our method starts by formulating an integer optimization problem that maximizes the rates that the Quantize-Map-and-Forward relaying protocol can achieve over a selected subnetwork; we then relax the integer constraints to obtain a non-linear optimization over reals. For diamond networks, we prove that this optimization over reals is concave while for general layered networks we give empirical demonstrations of near-concavity, paving the way for efficient algorithms to solve the relaxed problem. We then round the relaxed solution to select a specific subnetwork. Simulations using off-the-shelf non-linear optimization algorithms demonstrate excellent performance with respect to the true integer optimum for both diamond networks as well as multi-layered networks. Even with these non-customized algorithms, significant time savings are observed vis-a-vis exhaustive integer optimization.