Dynamic consolidation of virtual machines (VMs) in a cloud data center can be used to minimize
power consumption. Beloglazov et al. have proposed the MM (Minimization of Migrations) heuristic for
selecting the VMs to migrate from under- or over-utilized hosts, as well as the MBFD (Modified Best
Fit Decreasing) heuristic for deciding the placement of the migrated VMs. According to their simulation
results, these heuristics work very well in practice. In this paper, we investigate what performance
guarantees can be rigorously proven for the heuristics. In particular, we establish that MM is optimal
with respect to the number of selected VMs of an over-utilized host and it is a 1.5-approximation with
respect to the decrease in utilization. On the other hand, we show that the result of MBFD can be
arbitrarily far from the optimum. Moreover, we show that even if both MM and MBFD deliver optimal
results, their combination does not necessarily result in optimal VM consolidation, but approximation
results can be proven under suitable technical conditions. To the best of our knowledge, these are the
first rigorously proven results on the effectiveness of also practically useful heuristic algorithms for the
VM consolidation problem
