Barely Random Algorithms for Multiprocessor Scheduling

We present randomized online algorithms for scheduling on m = 3; : : : ; 7 machines. For two machines, a randomized algorithm achieving a competitive ratio of 4 3 was found by Bartal, Fiat, Karloo and Vohra 3]. These same authors show a matching lower bound. Seiden has presented a randomized algorithm which achieves competitive ratios of 1. A barely random algorithm is one which a distribution over a constant number of deterministic strategies. Neither of the algorithms of Bartal et al. or Seiden are barely random. In fact, both of these algorithms potentially make a random choice for each job scheduled. Further, both algorithms use (nm) variables, and use a total of (n 2 log m) time, where n is the length of the job sequence. Ben-David, Dichterman, Noga and Seiden have recently shown how to modify the two machine algorithm of Bartal et al. so that it is barely random 5]. We present a similar adaptation for Seiden's algorithm for m 3 machines. Our algorithms only use O(m) variables, and take a total of O(n log m) time. We accomplish this with only a slight increase in competitive ratio. Further, the changes we make to the algorithm simplify the analysis. For m = 3; 4 we note an improvement in the competitive ratio due to the use of a constraint overlooked in 11]. The new upper bounds for m = 3; 4 are 1:53727 and 1:65669, respectively.