The Santa Claus Problem in a Probabilistic Setting

In this paper we consider the Santa Claus problem. Santa Claus has n presents, which he wants to give to m children. Santa’s objective is to distribute the presents such that in the end the least happy child is as happy as possible. 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. Here we study a version of that problem with the additional restriction that Santa Claus has only a guess of how valuable a present is for a child, i.e., the value of a present respectively the processing time in the scheduling terminology, is a random variable. The scheduler is given the expectation λi of the processing time of the ith job for 1 ≤ i ≤ n, but the actual duration is not known in advance. Based on this information only the jobs have to be scheduled. In this setting we evaluate the performance of an algorithm using the competitive ratio. More precisely, we determine the value of the expected competitive ratio E [ Opt Alg ] , as a function of the λi, where Opt is the optimal algorithm that knows the realizations of the random variables in advance. In particular, we show that there is a critical value ρ such that if the expected values of the processing times deviate by less than a multiplicative factor of ρ from each other, then there exists an algorithm with expected competitive ratio arbitrarily close to one, i.e., Santa Claus can perform in expectation almost as good as an algorithm who knows the actual processing times in advance. On the other hand, if the expected values of the processing times deviate much from each other, then the expected performance can become arbitrarily bad for all algorithms. Our algorithm is nearly optimal also in this case.