The performance of distributed applications often critically depends on the interconnecting network or more specifically on its throughput: how fast data can be carried across a network. Over the last years, great progress has been made in understanding demand-oblivious throughput: how fast a given demand matrix describing pairwise communication requirements can be served on a given network. However, surprisingly little is known today about the achievable demand-aware throughput: the throughput on a network topology which can be optimized toward the demand. Such demand-aware networks have recently gained popularity in datacenters and are enabled by emerging reconfigurable optical technologies. In this paper, we are interested in both the achievable demand-aware throughput bounds as well as in the computational complexity of finding a throughput-optimizing network topology. We take a systematic approach and investigate four variants of demand-aware throughput: we analyze, and derive bounds for, two definitions of throughput, the classic throughput usually considered in the literature, and a new generalized definition which we call weak throughput; for each of them, we consider two routing models, a direct one, where demand can only be served on a single hop, and a general one, where multi-hop routing is allowed. Our main result is a separation result which solves an open problem in the literature about the classic throughput definition, showing that demand-aware topologies can outperform demand-oblivious topologies even in the worst case: the demand-aware throughput asymptotically approaches at least 5/8, while it is known that the demand-oblivious throughput is n/(2n-1), which is roughly 1/2. In terms of computational complexity, we show that computing the demand-aware weak throughput is NP-hard, but computing the demand-aware (weak) direct throughput is polynomial-time solvable.

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