Approximating Scheduling Problems in Parallel

We show how to approximate in NC the problem of Scheduling Unrelated Parallel Machines, for a xed number of machines. We develop a (2 + ")-approximate parallel algorithm for the problem. Our approach shows how to relate the linear program obtained by relaxing the integer programming formulation of the problem with a linear program formulation that is positive and in the packing/covering form. The relationship established enables us to transfer approximate fractional solutions from the later formulation that is known to be approximable in NC. Then, we show how to obtain an integer approximate solution, i.e. a schedule, from the fractional one, using the randomized rounding technique. Finally, we show that the same technique can be applied to the General Assignment Problem of xed number of machines and a given makespan T , thus yielding a schedule whose cost is at most (2+") times the minimum cost and has makespan at most 2T.