On the Asymptotic Worst Case Behavior of Harmonic Fit

In the parametric bin packing problem we must pack a list of items with size smaller than or equal to 1rr in a minimal number of unit-capacity bins. Among the Ž . approximation algorithms, the class of Harmonic Fit algorithms HF plays an M Ž Ž . . important role. Lee and Lee J. Assoc. Comput. Mach. 32 1985 , 562]572 and Ž Ž . . Galambos Ann. Unï . Sci. Budapest Sect. Comput. 9 1988 , 121]126 provide upper bounds for the asymptotic worst case ratio of HF and show tightness for M certain values of the parameter M. In this paper we provide worst case examples that meet the known upper bound for additional values of M, and we show that for remaining values of M the known upper bound is not tight. For the classical bin 12 Ž . packing problem r s 1 , we prove an asymptotic worst case ratio of for the case 7 M s 4 and 1.7 for the case M s 5. We give improved lower bounds for some interesting cases that are left open. Q 1996 Academic Press, Inc.