Pareto approximations for the bicriteria scheduling problem

Summary form only given. We consider the 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 fixed integer k, A_k is (2m-k/m-k+1,m+k-1/k)-competitive. Then we prove that, for m=2 and m=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 A₃ε ℱ₅is optimal. Finally, we extend our results to the general d-dimensional case with corresponding applications to the vector scheduling problem.