Scheduling identical jobs on uniform parallel machines

We address the problem of scheduling n identical jobs on m uniform parallel machines to optimize scheduling criteria that are nondecreasing in the job completion times. It is well known that this can be formulated as a linear assignment problem, and subsequently solved in O ( n ³) time. We give a more concise formulation for minsum criteria, and show that general minmax criteria can be minimized in O ( n ²) time. We present faster algorithms, requiring only O ( n + m log m ) time for minimizing makespan and total completion time, O ( n log n ) time for minimizing total weighted completion time, maximum lateness, total tardiness and the weighted number of tardy jobs, and O ( n log² n ) time for maximum weighted tardiness. In the case of release dates, we propose an O ( n log n ) algorithm for minimizing makespan, and an O ( mn ^2m+1) time dynamic programming algorithm for minimizing total completion time.