Bin Packing under Random-Order: Breaking the Barrier of 3/2

TL;DR

This work analyzes Best-Fit for online bin packing under the random-order model and achieves a notable breakthrough by proving a strictly sub-3/2 random-order ratio, namely $RR_{\mathrm{BF}}^{\infty} \le 3/2 - \varepsilon$ for some constant $\varepsilon>0$, while simultaneously improving the known lower bound to $1.144$. The authors introduce novel gadgets (e.g., $S$-triplets and fitting $ML/SL$ triplets) and weight-function techniques to structure the probabilistic analysis around a random permutation time $t_{\sigma}$, performing a careful case split that ensures many bins are either highly loaded or well matched to the optimum. A separate, computer-assisted construction based on an i.i.d. model yields a rigorous lower bound exceeding $1.144$, using a seven-item-type distribution and a 357-state Markov chain to bound BF’s behavior. Together, these results close much of the gap toward the conjectured ratio near $1.15$ and advance understanding of BF’s performance in realistically random inputs. The work also highlights the potential of gadgets and weight-function methodologies in random-order models for other packing and scheduling problems, with implications for offline packing heuristics and related online algorithms.

Abstract

Best-Fit is one of the most prominent and practically used algorithms for the bin packing problem, where a set of items with associated sizes needs to be packed in the minimum number of unit-capacity bins. Kenyon [SODA '96] studied online bin packing under random-order arrival, where the adversary chooses the list of items, but the items arrive one by one according to an arrival order drawn uniformly randomly from the set of all permutations of the items. Kenyon's seminal result established an upper bound of $1.5$ and a lower bound of $1.08$ on the random-order ratio of Best-Fit, and it was conjectured that the true ratio is $\approx 1.15$. The conjecture, if true, will also imply that Best-Fit (on randomly permuted input) has the best performance guarantee among all the widely-used simple algorithms for (offline) bin packing. This conjecture has remained one of the major open problems in the area, as highlighted in the recent survey on random-order models by Gupta and Singla [Beyond the Worst-Case Analysis of Algorithms '20]. Recently, Albers et al. [Algorithmica '21] improved the upper bound to $1.25$ for the special case when all the item sizes are greater than $1/3$, and they improve the lower bound to $1.1$. Ayyadevara et al. [ICALP '22] obtained an improved result for the special case when all the item sizes lie in $(1/4, 1/2]$, which corresponds to the $3$-partition problem. The upper bound of $3/2$ for the general case, however, has remained unimproved. In this paper, we make the first progress towards the conjecture, by showing that Best-Fit achieves a random-order ratio of at most $1.5 - \varepsilon$, for a small constant $\varepsilon>0$. Furthermore, we establish an improved lower bound of $1.144$ on the random-order ratio of Best-Fit, nearly reaching the conjectured ratio.

Figures (4)

• Figure 1: The overview of our case analysis. For brevity, we write $\mathrm{Opt}_1$ instead of $\mathrm{Opt}(I_\sigma(t_\sigma+1,n))$, $\mathrm{Opt}'_1$ instead of $\mathrm{Opt}(I'_\sigma(t_\sigma+1,n))$ (here $I'_\sigma(t_\sigma+1,n)$ denotes the list $I_\sigma(t_\sigma+1,n)$ after removing the small and tiny items), and $\#LM$ instead of "number of $LM$-bins".
• Figure 2: The Markov chain of the instance used by AlbersKL21 to prove a lower bound of $1.1$ on the random-order ratio of Best-Fit. The transition probabilities are $p=0.6,q=0.4$. The bold transitions indicate those that open a new bin.
• Figure 3: The items $y_1^2\le y_2^2\le y_3^2\le y_4^2$ denote the items of rank $2$ in the collection $C_3$. The items $x_1^2,x_2^2,x_3^2,x_4^2$ denote the corresponding master items.
• Figure 4: The bipartite graph $G_\Gamma$ when $x=4$. Every edge corresponds to a fitting $PQ$ pair. As a side note, the converse may not be true.

Theorems & Definitions (74)

• Theorem 1
• Theorem 2
• Lemma 2.1: Folklore
• proof
• Lemma 3.1: DBLP:conf/soda/Kenyon96
• Lemma 3.2: DBLP:conf/soda/Kenyon96
• Lemma 3.3
• proof : Proof Sketch
• Proposition 3.4
• Lemma 3.5