An Optimal Online Algorithm for Bounded Space Variable-Sized Bin Packing

An online algorithm for variable-sized bin packing, based on the Harmonic algorithm of Lee and Lee,[J. ACM, 32 (1985), pp. 562--572], is investigated. This algorithm was proposed by Csirik, [Acta Inform., 26 (1989), pp. 697--709], who proved that for all sets of bin sizes, 1.69103 upper bounds its performance ratio. The upper bound is improved in the sense that we give a method of calculating the performance ratio to any accuracy for any set of bin sizes. Further, it is shown that the algorithm is optimal among those which use bounded space. An interesting feature of the analysis is that, although it is shown that our algorithm achieves a performance ratio arbitrarily close to the optimum value, it is not known precisely what that value is. The case where bins of capacity 1 and $\alpha \in (0,1)$ are used is studied in greater detail. It is shown that among algorithms which are allowed to choose $\alpha$, the optimal performance ratio lies in [1.37530,1.37532].