A 4-Space Bounded Approximation Algorithm for Online Bin Packing Problem

Abstract

AbstractIn this work, we study the well-known online bin packing problem and propose a new approximation algorithm under the restriction of linear time and constant bounded space. In this setting, Lee and Lee in 1985 [12] introduced an algorithm called Harmonic that achieves the asymptotic approximation ratio (AAR) of \(T_\infty \approx 1.69\), and showed that no O(1)-space algorithm can obtain competitive ratio better than \(T_\infty \). Zhang et al. in 2000 [15] presented a linear time constant bounded space (number of bins kept during the execution of the algorithm is constant) online algorithm and proved that the absolute worst-case ratio is \(\frac{7}{4} = 1.75\). We extend Zhang et al. ’s work to consider three types of bins, and show that our proposed algorithm uses no more than \(\lceil \frac{55}{32}\cdot \textrm{OPT}\rceil \) bins. We first prove that the asymptotic approximation ratio is at most \(\frac{55}{32} = 1.71875\) by using the weight function, then we show that the additive term could be reduced from 3 to less than 1 if post-processing repacking is allowed. In the end, we provide a worst-case analysis and prove that the upper bound of \(\frac{55}{32}\) is tight.KeywordsBin packingOnline algorithmBounded-spaceCompetitive ratioWorse case analysis