A Cut-and-solve Algorithm for Virtual Machine Consolidation Problem

TL;DR

This paper introduces a new compact MILP formulation and a cut-and-solve algorithm for the virtual machine consolidation problem, significantly improving solution efficiency and scalability for large-scale instances.

Contribution

It presents a novel MILP formulation and a specialized cut-and-solve algorithm that outperform existing methods in solving VMCP efficiently and optimally.

Findings

The new formulation is more compact and scalable.

The cut-and-solve algorithm outperforms standard MILP solvers.

The approach achieves optimal solutions faster for large problems.

Abstract

The virtual machine consolidation problem (VMCP) attempts to determine which servers to be activated, how to allocate virtual machines (VMs) to the activated servers, and how to migrate VMs among servers such that the summation of activated, allocation, and migration costs is minimized subject to the resource constraints of the servers and other practical constraints. In this paper, we first propose a new mixed integer linear programming (MILP) formulation for the VMCP. We show that compared with existing formulations, the proposed formulation is much more compact in terms of smaller numbers of variables or constraints, which makes it suitable for solving large-scale problems. We then develop a cut-and-solve (C&S) algorithm, a tree search algorithm to efficiently solve the VMCP to optimality. The proposed C&S algorithm is based on a novel relaxation of the VMCP that provides a stronger lower bound than the natural continuous relaxation of the VMCP, making a smaller search tree. By extensive computational experiments, we show that (i) the proposed formulation significantly outperforms existing formulations in terms of solution efficiency; and (ii) compared with standard MILP solvers, the proposed C&S algorithm is much more efficient.