Influence of Variables Encoding and Symmetry Breaking on the Performance of Optimization Modulo Theories Tools Applied to Cloud Resource Selection

Abstract

The problem of Cloud resource provisioning for component-based applications is very important. It consists in the allocation of virtual machines (VMs) from various Cloud Providers (CPs), to a set of applications such that the constraints induced by the inter- actions between components and by the components hardware/software requirements are satisfied and the performance objectives are optimized (e.g. costs are minimized). It can be formulated as a constrained optimization problem and tackled by state-of-the-art optimization modulo theories (OMT) tools. The performance of the OMT tools is highly dependent on the way the problem is formalized as this determines the size of the search space. In the case when the number of VMs offers is large, a naive encoding which does not exploit the symmetries of the underlying problem leads to a huge search space making the optimization problem intractable. We overcame this issue by reducing the search space by using: (1) a heuristic which exploits the particularities of the application by detect- ing cliques in the conflict graph of the application components fixing all components of the clique with the largest number of component instances, and (2) a lex-leader method for breaking variable symmetry where the canonical solution fulfills an order based on either the number of components deployed on VMs, or on the VMs price. As the result, the running time of the optimization problem improves considerably and the optimization problem scales up to hundreds of VM offers. We also observed that by combining the heuristic with the lex-leader method we obtained better computational results than by using them separately, suggesting the fact that symmetry breaking constraints have the advantage of interacting well with the search heuristic being used.

Scalable Optimal Deployment in the Cloud of Component-based Applications using Optimization Modulo Theory, Mathematical Programming and Symmetry Breaking

“…As a result the scalability of the used optimizers increased, most notable, by at least 2 orders of magnitude in the case of the OMT solver. This paper extends our previous work [12,13,14] in the following aspects:…”

“…The estimated number of VMs reported on the second column has been obtained using the method described in Section 4.1. The list of offers was crawled from the Amazon site 13 . Each list of VM offers covers the main instance types, for example, small, medium, large.…”

“…For instance, in the case of a Secure Web Container Service (Section 2.1), the problem corresponding to 5 components and to a prior estimation of the number of VMs equal to 6, a solution can be (1), respectively (2). [12,12,13,13,15,0].…”