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30 János Balogh, József Békési, Gábor Galambos, and Mihály Csaba Markótratio (APR) R A of the algorithm A is defined asR A := lim sup k→∞ R A (k). (2)The best known lower bound for the AP R of any on-line bin packing algorithm A is 1.54014(given by Van Vliet [17]), while for the current best algorithm has an APR of 1.58889 (Seiden,[16]).The general semi-on-line bin packing problem with repacking was studied by Gambosi etal. [6]. Here the expression ‘general’ means that the number of repackable elements per stepis not restricted (bounded) in a strict way by a constant. They gave an O(n) time algorithmwith an AP R of 1.5 and an O(n log n) time algorithm with an AP R of 4 3. The latter algorithmwas improved to an algorithm with APR of 1.25 by Ivkovič and Lloyd [12]. Note, that none ofthe above mentioned algorithms repack constant number of elements in a strict sense, becausethey define the cost of repacking a bundle of small items as a constant.The only lower bound for this problem is proved by Ivkovič and Lloyd. This bound is 4 3 . Intheir lower bound construction the repacking of a constant number of items is allowed afterthe arrival of each new item. The 4 3lower bound is proved for fully dynamic bin packing withrestricted repacking. Dynamic bin packing means that in each step not only the insertion of thearrived elements (Insert operation) is allowed, but in any step one element can be deleted(Delete operation) instead of Insert. The fully dynamic bin packing is such a version of the dynamicbin packing when repacking is allowed. Although the 4 3lower bound was constructedfor this particular fully dynamic bin packing problem, the model can be easily applied for thesimilar classical (i.e. repacking, but not dynamic) semi-on-line bin-packing problem. For moredetails, see the survey of Csirik and Woeginger [3]).A similar problem class called batched bin packing is defined by Gutin et al. [9]: in this casethe items become available in batches, and each batch must be packed before the next batcharrives. By defining and solving a related nonlinear optimization problem we will generalizethe solution method of [9] for deriving lower bounds.In this paper we improve the lower bound 4/3 for that of the variant of the on-line binpacking problem when in each step the repacking of constant number of elements is allowed.The same construction can be used for deriving the same lower bound for the similar versionof the fully dynamic bin packing problem. Improved lower bounds will be obtained by solvinga nonlinear optimization problem. The solution of the special cases of the same nonlinearoptimization problem is interesting as well, because their solutions answer some questionsraised in [9] for the batched bin packing problem, namely, for the two batched bin packingproblem, when the number of different item sizes is at most p. In [9] it is shown that the lowerbound r(p) is the best possible in the case of p = 2, but it was not clear whether the boundsr(p) given for cases of p ≥ 3 might be the best possible bounds. Here we are answering thequestion regarding the optimality of the given bound for these values. We note that the lowerbounds for these special cases are valid for the above mentioned two problems, too.2. Constructing the lower bounds by a linear and a nonlinearoptimization problemThe following theorem is proved in [1]:Theorem 1. [1] Let k ≥ 1 and c ≥ 1 be arbitrary integers and x 1 , x 2 , . . . , x k , ( 1 2 ≤ x 1 < x 2