One of the important problems for datacenter resource management is to place virtual machines (VMs) to physical machines (PMs) such that certain cost, profit or performance objective is optimized, subject to various constraints. In this paper, we consider an interesting and difficult VM placement problem with disk anti-colocation constraints: a VM's virtual disks should be spread out across the physical disks of its assigned PM. For solutions, we use the mixed integer programming (MIP) formulations and algorithms. However, a challenge is the potentially long computation time of the MIP algorithms. In this paper, we explore how reformulation of the problem can help to reduce the computation time. We develop two reformulations, by redefining the variables, for our VM placement problem and evaluate the computation time of all three formulations. We show that they have vastly different computation time. All three formulations can be useful, but for different problem instances. They all should be kept in the toolbox for tackling the problem. Out of the three, formulation COMB is especially flexible and versatile, and it can solve large problem instances.

中文翻译：

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探索具有磁盘反托管约束的虚拟机放置的混合整数规划重构

数据中心资源管理的重要问题之一是将虚拟机 (VM) 放置到物理机 (PM) 上，以便优化某些成本、利润或性能目标，但受到各种限制。在本文中，我们考虑了一个有趣且困难的 VM 放置问题，该问题具有磁盘反托管约束：VM 的虚拟磁盘应该分布在其分配的 PM 的物理磁盘上。对于解决方案，我们使用混合整数规划 (MIP) 公式和算法。然而，一个挑战是 MIP 算法潜在的长计算时间。在本文中，我们探讨了问题的重新表述如何有助于减少计算时间。我们通过重新定义变量，为我们的 VM 放置问题开发了两个重新公式，并评估了所有三个公式的计算时间。我们表明它们具有截然不同的计算时间。所有三种公式都可能有用，但适用于不同的问题实例。它们都应该保存在工具箱中以解决问题。在这三者中，COMB 公式特别灵活和通用，它可以解决大型问题实例。