On-line load balancing made simple: Greedy strikes back

We provide a new approach to the on-line load balancing problem in the case of restricted assignment of temporary weighted tasks. The approach is very general and allows us to derive on-line algorithms whose competitive ratio is characterized by some combinatorial properties of the underlying graph G representing the problem: in particular, the approach consists in applying the greedy algorithm to a suitably constructed subgraph of G. In the paper, we prove the NP-hardness of the problem of computing an optimal or even a c-approximate subgraph, for some constant c > 1. Nevertheless, we show that, for several interesting problems, we can easily compute a subgraph yielding an optimal on-line algorithm. As an example, the effectiveness of this approach is shown by the hierarchical server model introduced by Bar-Noy et al. (2001). In this case, our method yields simpler algorithms whose competitive ratio is at least as good as the existing ones. Moreover, the algorithm analysis turns out to be simpler. Finally, we give a sufficient condition for obtaining, in the general case, $O(√n)$-competitive algorithms with our technique: this condition holds in the case of several problems for which a $\Omega( √n)$ lower bound is known.