All-Around Near-Optimal Solutions for the Online Bin Packing Problem

Abstract

In this paper we present the first algorithm with optimal average-case and close-to-best known worstcase performance for the classic on-line problem of bin packing. It has long been observed that known bin packing algorithms with optimal average-case performance were not optimal in the worst-case sense. In particular First Fit and Best Fit had optimal average-case ratio of 1 but a worst-case competitive ratio of 1.7. The wasted space of First Fit and Best Fit for a uniform random sequence of length n is expected to be Θ(n 2/3 ) and Θ( √ n log 3/4 n), respectively. The competitive ratio can be improved to 1.691 using the Harmonic algorithm; further variations of this algorithm can push down the competitive ratio to 1.588. However, Harmonic and its variations have poor performance on average; in particular, Harmonic has average-case ratio of around 1.27. In this paper, first we introduce a simple algorithm which we term Harmonic Match. This algorithm performs as well as Best Fit on average, i.e., it has an average-case ratio of 1 and expected wasted space of Θ( √ n log 3/4 n). Moreover, the competitive ratio of the algorithm is as good as Harmonic, i.e., it converges to 1.691 which is an improvement over 1.7 of Best Fit and First Fit. We also introduce a different algorithm, termed as Refined Harmonic Match, which achieves an improved competitive ratio of 1.636 while maintaining the good average-case performance of Harmonic Match and Best Fit. Finally, our extensive experimental evaluation of the studied bin packing algorithms shows that our proposed algorithms have comparable average-case performance with Best Fit and First Fit, and this holds also for sequences that follow distributions other than the uniform distribution.

Online Bin Packing with Predictions

“…The currently best algorithm is the Advanced Harmonic (AH) algorithm proposed by Balogh, Békési, Dósa, Epstein, and Levin (2018), which has a competitive ratio of 1.57829, whereas the best-known lower bound on the competitive ratio is 1.54278, as shown by Balogh, Békési, Dósa, Epstein, and Levin (2021). However, one should note that the Harmonic family of algorithms are designed specifically for improving the competitive ratio, and their typical performance is inferior to heuristics that are widely used in practice such as FIRSTFIT and BESTFIT, as discussed in Kamali and López-Ortiz (2015a).…”

Online Bin Packing with Predictions

“…It is worth noting that algorithms such as the one of [7] belong in a class that is tailored to worst-case competitive analysis (namely the class of harmonic-based algorithms) and do not tend to perform well on typical instances [22]. For this reason, simple algorithms such as FIRSTFIT and BESTFIT are preferred in practice, since they have a much better average-case performance at the expense of a somewhat inferior worst-case performance [13].…”

“…First, inputs generated from such simple distributions are often unrealistic and do not capture typical applications of bin packing such as resource allocation [17]. Second, in what concerns online algorithms, simple algorithms such as FIRSTFIT and BESTFIT are very close to optimal for input sequences generated from uniform distributions [13] and very often outperform, in practice, many online algorithms of better competitive ratio [22].…”

Maximizing space utilization in unit-load warehouses.

“…Another approach to evaluate the performance of online algorithms is the average-case analysis that focuses on a typical scenario (Shor, 1991), where First-Fit and Best-fit present the best performance (Kamali and López-Ortiz, 2015)-the asymptotic average-case performance ratio of First-fit and Best-fit is 1 while that of the Harmonic is 1.29 (Shor, 1986).…”

“…The most promising approaches are Best-fit (Johnson, 1974), Harmonic-k (Lee and Lee, 1985), Sum of Squares (Csirik et al, 2006), Lagrangian-based (Gupta and Radovanovic, 2015), and CHAMP (Asta et al, 2016). Nevertheless, Best-fit seems to have the best performance in practice (Ghaderi et al, 2014;Rajagopal, 2016;Kamali and López-Ortiz, 2015). Therefore, we define Best-fit as the dispatching policy A of our simulation model S. Specifically, to put a pallet of height h away, A identifies all the rack-bays where it fits.…”