Performance Ratios for the Karmarkar-Karp Differencing Method

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 due to Karmarkar and Karp outperforms other existing polynomial-time approximation algorithms from an average-case perspective. For m ≥ 3, its worst-case performance has remained a challenging open problem. We show that its performance ratio is bounded between 43 − 1(3(m-1)) and 43−13m. We also analyze the performance ratio if in addition to the number of machines, the number of tasks n is fixed as well.