Online Scheduling Revisited Rudolf Fleischer 1 and Michaela Wahl 2 1 University of Waterloo, Department of Computer Science email: rudolf@uwaterloo.ca 2 Max-Planck-Institut f¨ ur Informatik, Saarbr¨ ucken email: altherr@mpi-sb.mpg.de Abstract. We present a new online algorithm, MR, for non-preemptive scheduling of jobs with known processing times on m identical machines which beats the best previous algorithm for m ≥ 64. For m →∞ its competitive ratio approaches 1+  1+ln 2 2 < 1.9201. 1 Introduction Scheduling problems are of great practical interest. However, even some of the simplest variants of the problem are not fully understood. In this paper, we study the classical problem of scheduling jobs online on m identical machines without preemption, i.e., the jobs arrive one at a time with known processing times and must immediately be scheduled on one of the machines, without knowledge of what jobs will come afterwards, or how many jobs are still to come. The goal is to achieve a small makespan which is the total processing time of all jobs scheduled on the most loaded machine. Since the jobs must be scheduled online we cannot expect to achieve the minimum makespan whose computation would require a priori knowledge of all jobs (even then computing the minimum makespan is difficult, i.e., NP-hard [12]). The quality of an online algorithm is therefore measured by how close it comes to that optimum [4,10]. It is said to be c-competitive if its makespan is at most c times the optimal makespan for all possible job sequences. Graham [14] showed some 30 years ago that the List algorithm which always puts the next job on the least loaded machine is exactly (2 - 1 m )-competitive. Only much later were better algorithms designed. A series of papers improved the upper bound on the competitive ratio of the online scheduling problem first to (2 - 1 m - ǫ m )[11,5], then to 1.986 for m ≥ 70[2], then to 1.945 [16], and finally to 1.923 [1]. On the other hand, the lower bound for the problem was raised in similarly small steps: From 1.707 for m ≥ 4[9], to 1.837 for m ≥ 3454 [3], to 1.852 for m ≥ 80 [1], and finally to 1.85358 for m ≥ 80 [13]. Better bounds are known for a few special cases. For m = 2 and m = 3, the lower bound is (2 - 1 m )[9], i.e., List is optimal. And for m = 4, a 1.733- competitive algorithm is known [7]. The best lower bound for randomized al- gorithms is e e-1 ≈ 1.58 for large m [6,18], and at least for m ≤ 5 randomized algorithms can beat the best deterministic lower bound [17]. For scheduling with M. Paterson (Ed.): ESA 2000, LNCS 1879, pp. 202–210, 2000. c  Springer-Verlag Berlin Heidelberg 2000