A Generalized Multi-organization Scheduling on Unrelated Parallel Machines

We consider the parallel computing environment where m organizations provide machines and several jobs to be executed. While cooperation of organizations is required to minimize the global makespan, each organization also expects the faster completion of its own jobs primarily and thus it is not necessarily cooperative. To handle the situations, we formulate the alpha-cooperative multi-organization scheduling problem (alpha-MOSP), where alpha >= 1 is a parameter representing the degree of cooperativeness. alpha-MOSP minimizes the makespan under the cooperation constraint that each organization does not allow the completion time of its own jobs to be delayed alpha times of that in the case where those jobs are executed by itself. In this paper, we aim to reveal the relation between the makespan and the degree of cooperativeness. First, we investigate the relation between alpha and the quality of the global makespan. For alpha = 1 (i.e., each organization never sacrifices its completion time), we show an instance where the cooperation constraint degrades the optimal makespan by m times. In contrast, for alpha > 1, we can construct an algorithm transforming any unconstrained schedule to one satisfying the cooperation constraint. This algorithm bounds the degradation ratio by alpha/(alpha - 1), which implies that weak cooperation improves the makespan dramatically. Second, we study the complexity of alpha-MOSP. We show its strongly NP-hardness and inapproximability for the approximation factor less than max{(alpha 1)/alpha,3/2}. We also show the hardness of transformation: Even if an optimal schedule under no cooperation constraint is given, no polynomial-time algorithm finds an optimal schedule for a-MOSP. This result is a witness for inexistence of general polynomial-time transformation algorithms that preserve the approximation ratio.