In this paper 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 \le k \le m\}such that, for each fixed integer k, A_k is ({{2m - k} \over {m - k + 1}},{{m + k - 1} \over 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 \over 4} for m = 4, r = {6 \over 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 \over 3}, {7 \over 3}) of A_3 \in F_5 is optimal.

Finally, we extend our results to the general d-dimensional case with corresponding applications to the Vector Scheduling problem.