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Multiprocessor platforms have been widely adopted to accommodate the increasing computation requirement of modern applications. Partitioned scheduling (or packing) has been widely exploited by partitioning real-time tasks onto processors to meet the timing constraints, which has been shown to be NP-complete in the strong sense. This paper studies the approximation of partitioned scheduling by exploiting resource augmentation with (1) speeding up or (2) allocating more processors. When adopting speeding up to meet timing constraints, we provide a polynomial-time approximation scheme (PTAS) to derive near-optimal solutions only with the assumption that the ratio of the maximum relative deadline to the minimum relative deadline is a constant. The previously known PTAS for this problem imposes additional restrictions on the periods and the execution times of tasks. By removing these additional constraints, our scheme can be adopted for wider task sets. When considering the resource augmentation by allocating more processors, we show that there does not exist any asymptotic polynomial-time approximation scheme (APTAS) unless P=NP.