Tight Approximation for Scheduling Parallel Jobs on Identical Clusters

We consider the Multiple Cluster Scheduling Problem (MCSP), where the objective is to schedule n parallel rigid jobs on N identical clusters, minimizing the maximum completion time (makespan). MCSP is 2-inapproximable (unless P = NP), and several approximation algorithms have already been proposed. However, ratio 2 has only been reached by algorithms that use extremely costly and complex subroutines as ”black boxes” which are polynomial and yet impractical due to prohibitive constants. Our objective within this work is to determine a reasonable restriction of MCSP where the inapproximability lower bound could be tightened in almost linear time. Thus, we consider a restriction of MCSP where jobs do not require strictly more than half of the processors of a cluster, and we provide a 2-approximation running in O(log(nh_max)n(N + log(n))), where h_max is the processing time of the longest job. This approximation is the best possible, as this restriction (and even simpler ones) remains 2-inapproximable.