Multi-machine Scheduling with Variance Minimization

This paper addresses the problem of scheduling n jobs on m identical parallel machines so as to minimize the completion time variance. Properties of optimal solutions are derived first. Then, complexity results are obtained, which show that the problem is NP-complete in the strong sense when m is arbitrary, and NP-complete in the ordinary sense when m is fixed. Two algorithms are proposed. The first algorithm can generate an optimal solution in time O(n2”P”(P P,)“-‘/[m”(m l)!]*), w h ere P is the sum of all the processing times and P, is the sum of the first m largest processing times. The second algorithm can find a near-optimal solution in time O(nP”(P Pm)m-‘/mm(m l)!). It is further shown that the relative error of the near-optimal solution is guaranteed to approach zero at a rate O(np2) as n increases.