Online Load Balancing on Unrelated Machines with Startup Costs

Abstract

Motivated by applications in energy-efficient scheduling in data centers, Khuller, Li, and Saha introduced the {\em machine activation} problem as a generalization of the classical optimization problems of set cover and load balancing on unrelated machines. In this problem, a set of $n$ jobs have to be distributed among a set of $m$ (unrelated) machines, given the processing time of each job on each machine, where each machine has a startup cost. The goal is to produce a schedule of minimum total startup cost subject to a constraint $\bf L$ on its makespan. While Khuller {\em et al} considered the offline version of this problem, a typical scenario in scheduling is one where jobs arrive online and have to be assigned to a machine immediately on arrival. We give an $(O(\log (mn)\log m), O(\log m))$-competitive randomized online algorithm for this problem, i.e. the schedule produced by our algorithm has a makespan of $O({\bf L} \log m)$ with high probability, and a total expected startup cost of $O(\log (mn)\log m)$ times that of an optimal offline schedule with makespan $\bf L$. The competitive ratios of our algorithm are (almost) optimal. Our algorithms use the online primal dual framework introduced by Alon {\em et al} for the online set cover problem, and subsequently developed further by Buchbinder, Naor, and co-authors. To the best of our knowledge, all previous applications of this framework have been to linear programs (LPs) with either packing or covering constraints. One novelty of our application is that we use this framework for a mixed LP that has both covering and packing constraints. We hope that the algorithmic techniques developed in this paper to simultaneously handle packing and covering constraints will be useful for solving other online optimization problems as well.