Improved approximation algorithms for scheduling parallel jobs on identical clusters

The Multiple Cluster Scheduling Problem corresponds to minimizing the maximum completion time (makespan) of a set of n parallel rigid (and non-preemptive) jobs submitted to N   identical clusters. It cannot be approximated with a ratio better than 2 (unless P=NPP=NP). We present in this paper the methodology that encompasses several existing results [1] and [2]. We detail first how to apply it for obtaining a 52-approximation. Then, we use it to provide a new 73-approximation running in O(log⁡(nhmax)N(n+log⁡(n)))O(log⁡(nhmax)N(n+log⁡(n))), where hmaxhmax is the processing time of the longest job. Finally, we apply it to a restriction of the problem to jobs of limited size, leading to a 2-approximation which is the best possible ratio since the restriction remains 2-inapproximable.