Improved Bounds for the Online Scheduling Problem

Abstract

The problem considered here is the same as the one discussed in [G. Galambos and G. J. Woeginger, eds., SIAM J. Comput., 22 (1993), pp. 349--355]. It is an m-machine online scheduling problem in which we wish to minimize the competitive ratio for the makespan objective. In this paper, we show that $\sqrt{3}$ is a lower bound on this competitive ratio for m=4. In particular, we show how to force a lower bound of $\sqrt{3\,}-\epsilon $ for any positive $\epsilon $. This reduces the gap between the performance of known algorithms [S. Albers, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, ACM, New York, 1997, pp. 130--139] and the lower bound. The method used introduces an approach to building the task master's strategy.