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Infrastructure-as-a-Service (IaaS) providers need to offer richer services tobe competitive while optimizing their resource usage to keep costs down. Richerservice offerings include new resource request models involving bandwidthguarantees between virtual machines (VMs). Thus we consider the followingproblem: given a VM request graph (where nodes are VMs and edges representvirtual network connectivity between the VMs) and a real data center topology,find an allocation of VMs to servers that satisfies the bandwidth guaranteesfor every virtual network edge---which maps to a path in the physicalnetwork---and minimizes congestion of the network.  Previous work has shown that for arbitrary networks and requests, finding theoptimal embedding satisfying bandwidth requests is $\mathcal{NP}$-hard.However, in most data center architectures, the routing protocols employed arebased on a spanning tree of the physical network. In this paper, we prove thatthe problem remains $\mathcal{NP}$-hard even when the physical network topologyis restricted to be a tree, and the request graph topology is also restricted.We also present a dynamic programming algorithm for computing the optimalembedding in a tree network which runs in time $O(3^kn)$, where $n$ is thenumber of nodes in the physical topology and $k$ is the size of the requestgraph, which is well suited for practical requests which have small $k$. Suchrequests form a large class of web-service and enterprise workloads. Also, ifwe restrict the requests topology to a clique (all VMs connected to a virtualswitch with uniform bandwidth requirements), we show that the dynamicprogramming algorithm can be modified to output the minimum congestionembedding in time $O(k^2n)$.

Optimal bandwidth-aware VM allocation for Infrastructure-as-a-Service