We consider load balancing in the following setting. The on-line algorithm is allowed to use n machines, whereas the optimal off-line algorithm is limited to m machines, for some fixed m < n. We show that while the greedy algorithm has a competitive ratio which decays linearly in the inverse of n/m, the best on-line algorithm has a ratio which decays exponentially in n/m. Specifically, we give a deterministic algorithm with competitive ratio of 1 + 2−n/m(1−o(1)), and a lower bound of 1 + e−(n/m)(1+o(1)) on the competitive ratio of any randomized algorithm. We also consider the preemptive case. We show an on-line algorithm with a competitive ratio of 1 + e−(n/m)(1+o(1)). We show that the algorithm is optimal by proving a matching lower bound. We also consider the non-preemptive model with temporary tasks. We prove that for n = m + 1, the greedy algorithm is optimal. (It is not optimal for permanent tasks.)
