On-line machine covering

We consider the problem of scheduling a sequence of jobs to m parallel machines as to maximize the minimum load over the machines. This situation corresponds to a case that a system which consists of the m machines is alive (i.e. productive) only when all the machines are alive, and the system should be maintained alive as long as possible. It is well known that any on-llne deterministic algorithm for identical machines has a competitive ratio of at least m and that greedy is an m competitive algorithm. In contrast we design an on-line randomized algorithm which is Ö(√M) competitive and a matching lower bound of Ω(√M) for any online randomized algorithm. In the case where the jobs are polynomially related we design an optimal O(log m) competitive randomized algorithm and a matching tight lower bound for any on-line randomized algorithm. In fact, if F is the ratio between the largest job and the smallest job then our randomized algorithm is O(log F) competitive. A sub-problem that we solve which is interesting by its own is the problem where the value of the optimal algorithm is known in advance. Here we show a deterministic (constant) 2-1/m competitive algorithm. We also show that our algorithm is optimal for two, three and four machines and that no on-fine deterministic algorithm can achieve a better competitive ratio than 1.75 for m ≥ 4 machines. For related machines we show that there is no on-line algorithm, whose competitive ratio is a function of the number of machines. However, for the case where the value of the optimal assignment is known in advance, and for the case where jobs arrive in non increasing order, we show that the exact competitive ratio is m. We show a constant 2 competitive algorithm for the intersection of the above two cases, i.e. the value of the optimal assignment is known in advance and the jobs arrive in non increasing order.