Modern cloud infrastructure providers allow customers to rent computing capability in the form of a network of virtual machines (VMs) with bandwidth guarantees between pairs of VMs. Typical requests are in the form of a chain of VMs with an uplink bandwidth to the gateway node of the network (rooted path requests), and most data center architectures route network packets along a spanning tree of the physical network. VMs are instantiated inside servers which reside at the leaves of this network, leading to the following optimization problem: given a rooted tree network T and a set of rooted path requests, find an embedding of the requests that minimizes link congestion. Our main result is an algorithm that, given a rooted tree network T with n leaves and set of weighted rooted path requests, embeds a 1-ε fraction of the requests with congestion at most poly(logn, logθ,ε -1). OPT (approximation is necessary since the problem is NP-hard). Here OPT is the congestion of the optimal embedding and θ is the ratio of the maximum to minimum weights of the path requests. We also obtain an O(Hlogn/ε 2) approximation if node capacities can be augmented by a (1+ε) factor (here H is the height of the tree). Our algorithm applies a randomized rounding scheme based on Group Steiner Tree rounding to a novel LP relaxation of the set of subtrees of T with a given number of leaves that may be of independent interest. © 2012 Springer-Verlag.
CITATION STYLE
Dutta, D., Kapralov, M., Post, I., & Shinde, R. (2012). Embedding paths into trees: VM placement to minimize congestion. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7501 LNCS, pp. 431–442). https://doi.org/10.1007/978-3-642-33090-2_38
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