On the online bin packing problem
Journal of the ACM (JACM)
Bin packing with controllable item sizes
Information and Computation
Variable sized online interval coloring with bandwidth
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Online bin packing with cardinality constraints
ESA'05 Proceedings of the 13th annual European conference on Algorithms
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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].