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 < k <= m}, such that, for each fixedinteger k, A(k) is (2m-k/m-k+1, m+k-1/k)- competitive. Then we prove that, for m = 2 and 3, the tradeoffs on the competitive ratios realized by the algorithms in F-m correspond to the Pareto curve, that is they are all and only the optimal ones, while for m > 3 they give an r-approximation of the Pareto curve with r = 5/4 for m = 4, r = 6/5 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 (7/3, 7/3) of A(3) is an element of F-5 is optimal. Finally, we extend our results to the general d-dimensional case with corresponding applications to the Vector Scheduling problem. (C) 2005 Elsevier Inc. All rights reserved.