Fast near-optimal subnetwork selection in layered relay networks

We consider the problem of finding the largest capacity subnetwork of a given size of a layered Gaussian relay network. While the exact capacity of Gaussian relay networks is unknown in general, motivated by recent capacity approximations we use the information-theoretic cutset bound as a proxy for the true capacity of such networks. There are two challenges in efficiently selecting subnetworks of a Gaussian network. First, evaluating the cutset bound involves a minimization of a cut function over the exponentially many possible cuts of the network and therefore a greedy approach has exponential complexity. Second, even if the min-cut for each subnetwork can be evaluated efficiently, an exhaustive search over the possibly exponentially many subnetworks of a network has prohibitive complexity. Algorithms exploiting the submodularity property of the cut function have been proposed in the literature to address these challenges. Instead, in this paper, we develop algorithms for computing the min-cut of a layered network and selecting its largest capacity subnetwork which are based on the observation that the cut function of a layered network admits a line-structured factor graph representation. We demonstrate numerically that our algorithms exploiting the layered structure can be significantly more efficient than the earlier algorithms exploiting submodularity. Our findings suggest that while submodularity of the cut function holds in more generality independent of the topology of the network, in the case of layered networks, algorithms exploiting the layered structure of the cut function can be much more efficient.