Online Load Balancing on Unrelated Machines with Startup Costs

Motivated by applications in energy-efficient scheduling in data centers, Khuller, Li, and Saha introduced the machine activation problem as a generalization of the classical optimization problems of minimum set cover and minimum makespan scheduling on parallel 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. Additionally, each machine incurs a startup cost if at least one job is assigned to it. The goal is to produce a schedule of minimum total startup cost subject to a constraint L on its makespan. While Khuller 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) logm),O(logm))-competitive randomized online algorithm for this problem, i.e. the schedule produced by our algorithm has a makespan of O(L logm) with high probability, and a total expected startup cost of O(log(mn) logm) times that of an optimal offline schedule with makespan L. Our algorithm is almost optimal since it follows from previous results that the two approximation factors cannot be improved to o(logm logn) (under standard complexity assumptions) and o(logm) respectively. Our algorithms use the online primal dual framework introduced by Alon et al for the online set cover problem, and subsequently developed further by Buchbinder, Naor and co-authors in various papers. 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 combine the packing constraint with the objective function to design a potential function on the machines that is exponential in the current load of the machine and linear in the cost of the machine. Then, we create a dynamic order of machines based on this potential function and assign larger fractions of the job to machines that appear earlier in this order. This allocation is somewhat unusual in that the increase in load on a machine is inverse in the value of this potential function itself, i.e. inverse exponential in the current load on the machine. Finally, we show that we can round this fractional solution online using a randomized algorithm. 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. ∗Part of this work was done while the first author was visiting and the second author was an intern at Microsoft Research, Redmond, WA 98052. †Blatavnik School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel. Email: [email protected]. ‡Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Email: [email protected].