An O(m^2 log m)-Competitive Algorithm for Online Machine Minimization

We consider the online machine minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. Our main result is a general O(m² log m)-competitive algorithm for the preemptive online problem, where m is the optimal number of machines used in an offline solution. This is the first improvement on an O(log (pmax/pmin))-competitive algorithm by Phillips et al. (STOC 1997), which was to date the best known algorithm even when m=2. Our algorithm is O(1)-competitive for any m that is bounded by a constant. To develop the algorithm, we investigate two complementary special cases of the problem, namely, laminar instances and agreeable instances, for which we provide an O(log m)-competitive and an O(1)-competitive algorithm, respectively. Our O(1)-competitive algorithm for agreeable instances actually produces a non-preemptive schedule, which is of its own interest as there exists a strong lower bound of n, the number of jobs, for the general non-preemptive online machine minimization problem by Saha (FSTTCS 2013), which even holds for laminar instances.