Pareto approximations for the bicriteria scheduling problem

In this paper, we consider the online bicriteria version of the classical Graham's scheduling problem in which two cost measures must be simultaneously minimized. We present a parametric family of online algorithms F"m={A"k|1= 3 they give an r-approximation of the Pareto curve with r=54 for m=4, r=65 for m=5, r=1.186 for m=6 and so forth, with r always less than 1.295. Unfortunately, for m>3, obtaining Pareto curves is not trivial, as they would yield optimal algorithms for the single criterion case in correspondence of the extremal tradeoffs. However, the situation seems more promising for the intermediate cases. In fact, we prove that for 5 processors the tradeoff 73,73 of A"3@__ __F"5 is optimal. Finally, we extend our results to the general d-dimensional case with corresponding applications to the Vector Scheduling problem.