Abstract
In this paper, we analyze the average-case performance of the Modified Harmonic algorithm for bin packing. We first analyze the average-case performance for arbitrary distribution of item sizes over (0, 1]. This analysis is based on the following result. Let f1 and f2 be two linear combinations of random variables {ni} n=1 k , where the Nis have a joint multinomial distribution for each \(n = \sum\limits_{i = 1}^k {N_i }\). Let E(f1)≠0, and E(f2)↮0. Then \(\mathop {\lim }\limits_{n \to \infty }\)E(max(f1, f2))/n=\(\mathop {\lim }\limits_{n \to \infty }\)max(E(f1), E(f2))/n. We then consider the special case when the item sizes are uniformly distributed over (0, 1], and obtain optimal values for the parameters of the algorithm. For these values of the parameters, the average-case performance ratio is less than 1.19. This compares well with the performance ratio 1.2865... of the Harmonic algorithm.
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