Maximizing the Minimum Load for Random Processing Times

Abstract

In this article, we consider a stochastic variant of the so-called Santa Claus problem. The Santa Claus problem is equivalent to the problem of scheduling a set of n jobs on m parallel machines without preemption, so as to maximize the minimum load. We consider the identical machine version of this scheduling problem with the additional restriction that the scheduler has only a guess of the processing times; that is, the processing time of a job is a random variable . We show that there is a critical value ρ ( n,m ) such that if the duration of the jobs is exponentially distributed and the expected values deviate by less than a multiplicative factor of ρ ( n,m ) from each other, then a greedy algorithm has an expected competitive ratio arbitrarily close to one; that is, it performs in expectation almost as good as an algorithm that knows the actual values in advance . On the other hand, if the expected values deviate by more than a multiplicative factor of ρ ( n,m ), then the expected performance is arbitrarily bad for all algorithms.