In the data flow models of today's data center applications such as MapReduce, Spark and Dryad, multiple flows can comprise a coflow group semantically. Only completing all flows in a coflow is meaningful to an application. To optimize application performance, routing and scheduling must be jointly considered at the level of a coflow rather than individual flows. However, prior solutions have significant limitation: they only consider scheduling, which is insufficient. To this end, we present RAPIER, a coflow-aware network optimization framework that seamlessly integrates routing and scheduling for better application performance. Using a smallscale testbed implementation and large-scale simulations, we demonstrate that RAPIER significantly reduces the average coflow completion time (CCT) by up to 79.30% compared to the state-of-the-art scheduling-only solution, and it is readily implementable with existing commodity switches.

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Traffic engineering (TE) is a fundamental task in networking. Conventionally, traffic can take any path connecting the source and destination. Emerging technologies such as segment routing, however, use logical paths going through a predetermined set of middlepoints. Inspired by this, in this work we introduce the problem of node-constrained TE, where traffic must go through a set of middlepoints, and study its theoretical fundamentals. We show that the general node-constrained TE that constrains the traffic to take paths going through one or more middlepoints is NP-hard for directed graphs but strongly polynomial for undirected graphs, unveiling a profound dichotomy between the two cases. We additionally investigate the popular variant of node-constrained TE that uses shortest paths between middlepoints, and show that the problem can now be solved in weakly polynomial time for a fixed number of middlepoints. Yet if we constrain the end-to-end paths to be acyclic, the problem can become NP-hard. This explains why existing work focuses on the computationally tractable variant. An important application of our work concerns the computational complexity of flow centrality, first proposed in 1991 by Freeman et al. [21]: we show that it is NP-hard for directed but strongly polynomial for undirected graphs. Finally, we investigate the middlepoint selection problem in general node-constrained TE. We introduce group flow centrality as a solution concept for multi-commodity networks, study its complexity, and show that it is monotone but not submodular for both directed and undirected graphs. Our work provides a thorough theoretical treatment of nodeconstrained TE and its applications.

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