Randomized Approximation Schemes for Scheduling Unrelated Parallel Machines

The problem of Scheduling n Independent Jobs on m Unrelated Parallel Machines, when the number of machines m is xed, is considered. The standard problem of minimizing the makespan of the schedule (SUM) and the bicriteria problem of scheduling with bounded makespan and cost (SUMC) are addressed, and randomized fully linear time approximation schemes are shown for both of them. While matching the approximation guarantee and the complexity of the best known sequential results of Jansen and Porkolab ((12]), the proposed algorithms exhibit a signiicantly simpler and more general rounding scheme, especially for the bicriteria SUMC problem, and admit simple optimal work parallelizations 1 of O(log n){time complexity. The core of the algorithms, which also draw techniques from other related works ((12], 11], 1]), is an interesting new randomized rounding procedure, the Filtered Randomized Rounding (FRR). In the settings of the problems considered, FRR boosts the deviation bounds of the rounded linear packing constraints to any given constant ratio. Finally, the notion of poly{bottleneck combinatorial optimization problems is deened and used to build O(n log n log log n) time approximation schemes for two natural optimization versions of SUMC, that is minimizing the makespan when the cost of the schedule is bounded (SUMCoptT) and minimizing the cost when the makespan is bounded (SUMCoptC). These algorithms too, admit simple optimal work parallelizations. 1 Optimal work parallelization means that the parallel work (product of running time and number of processors) is equal to the sequential running time.