Abstract
Analysis of Algorithms In the extensible bin packing problem we are asked to pack a set of items into a given number of bins, each with an original size. However, the original bin sizes can be extended if necessary. The goal is to minimize the total size of the bins. We consider the problem with unequal (original) bin sizes and give the complete analysis on a list scheduling algorithm (LS). Namely we present tight bounds of LS for every collection of original bin sizes and every number of bins. We further show better on-line algorithms for the two-bin case and the three-bin case. Interestingly, it is proved that the on-line algorithms have better competitive ratios for unequal bins than for equal bins. Some variants of the problem are also discussed.
Highlights
We consider the following on-line extensible bin packing problem: there are m bins B1, B2, . . . , Bm with original bin sizes b1, b2, . . . , bm
We prove the competitive ratio of list scheduling algorithm (LS) for the two-bin case and present an improved on-line algorithm
We present an on-line algorithm to improve the upper bound in some cases
Summary
We consider the following on-line extensible bin packing problem: there are m bins B1, B2, . . . , Bm with original bin sizes b1, b2, . . . , bm. For the lower bound of the on-line problem, they showed that no heuristic can have a competitive ratio smaller than 7/6 by presenting an instance for the case that m = 2. Ye and Zhang [12] considered on-line scheduling on a small number m of bins They proved lower bounds for m = 3, 4 and gave the competitive ratios RHx (m) for m = 2, 3 and 4. We prove the competitive ratio of LS for the two-bin case and present an improved on-line algorithm. It implies that we only need to consider the instances in which the total size of items is at least the total original size of bins to prove an upper bound.
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