Optimal Online Algorithms for Multidimensional Packing Problems

Abstract

We solve an open problem in the literature by providing an online algorithm for multidimensional bin packing that uses only bounded space. To achieve this, we introduce a new technique for classifying the items to be packed. We show that our algorithm is optimal among bounded space algorithms for any dimension d > 1. Its asymptotic performance ratio is (Π ∞) d , where Π ∞ ≈ 1.691 is the asymptotic performance ratio of the one-dimensional algorithm Harmonic. A modified version of this algorithm for the case where all items are hypercubes is also shown to be optimal. Its asymptotic performance ratio is sublinear in d. Furthermore, we extend the techniques used in these algorithms to give optimal algorithms for online bounded space variable-sized packing and resource augmented packing.

On Variable-Sized Multidimensional Packing

“…For bounded space algorithms, a lower bound of (Π ∞ ) d is implied by [5]. A matching upper bound was shown in [7]. This was done by giving an extension of HARMONIC which uses only bounded space.…”

“…For the variable-sized packing, the method used in this paper generalizes and combines the methods used in [7] and [14]. The are some new ingredients that we use in this paper.…”

“…An important difference with the previous papers is that in the current problem, it is no longer immediately clear which bin size should be used for any given item. In [7] all bins have the same size. In [14] bins have one dimension and therefore a simple greedy choice gives the best option.…”

Online algorithms for 1-space bounded multi dimensional bin packing and hypercube packing

“…Csirik and Johnson (2001) presented an 1.7-competitive algorithm (K-Bounded Best Fit algorithms (BBF K )) for one dimensional bin packing using K active bins, where K ≥ 2. For multi-dimensional case, Epstein and van Stee (2005b) Chin et al (2012) proposed an 8.84-competitive packing strategy, then they further improved the upper bound to 5.155 (Zhang et al 2010), they also gave the lower bound 3 for 1-space bounded two dimensional bin packing.…”

Improved Online Algorithms for 1-Space Bounded 2-Dimensional Bin Packing

“…Epstein et al [8] gave a 1.69103 d -competitive algorithm using (2M − 1) d active bins, where M ≥ 10 is an integer such that M ≥ 1/(1 − (1 − ε) 1/(d+2) ) − 1, ε > 0 and d is the dimension of the bin packing problem. For the 1-space bounded variant, Fujita [12] gave an O((log log m) 2 )-competitive algorithm, where m is the width of the square bin and the size of each item is a × b, where a, b are integers no larger than m. He also proved that the competitive ratio for the 1-bounded space variant is at least 23/11.…”