Abstract
We consider the multiprocessor scheduling problem in which one must schedule n independent tasks nonpreemptively on m identical, parallel machines, such that the completion time of the last task is minimal. For this well-studied problem the Largest Differencing Method of Karmarkar and Karp outperforms other existing polynomial-time approximation algorithms from an average-case perspective. For m ≥ 3 the worst-case performance of the Largest Differencing Method has remained a challenging open problem. In this paper we show that the worst-case performance ratio is bounded between \( frac{4}{3}-\frac{1}{3(m-1)}$ and $\frac{4}{3}-\frac{1}{3m}\). For fixed m we establish further refined bounds in terms of n.
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