Near-optimal Algorithms for Stochastic Online Bin Packing

TL;DR

This work studies online bin packing under two stochastic models, presenting near-optimal algorithms. It introduces a meta-algorithm that converts any offline α-approximation with AAR α into an online algorithm with ECR α+ε in the i.i.d. setting, and shows that using an AFPTAS yields a near-ideal (1+ε)-competitive online algorithm. In the random-order model, the authors prove BF has ARR = 1 for item sizes above 1/3 and obtain a concrete ARR of 1.49107 for the challenging 3-Partition regime, indicating BF may beat the 3/2 barrier in practice. They also extend the framework to multidimensional online vector packing, achieving a (dα+εd)-competitive algorithm via a rounding reduction to 1D. Altogether, the results bridge offline-online gaps, provide near-optimal guarantees under stochastic inputs, and open avenues for further improvement in randomness-aware online packing problems.

Abstract

We study the online bin packing problem under two stochastic settings. In the bin packing problem, we are given n items with sizes in (0,1] and the goal is to pack them into the minimum number of unit-sized bins. First, we study bin packing under the i.i.d. model, where item sizes are sampled independently and identically from a distribution in (0,1]. Both the distribution and the total number of items are unknown. The items arrive one by one and their sizes are revealed upon their arrival and they must be packed immediately and irrevocably in bins of size 1. We provide a simple meta-algorithm that takes an offline $α$-asymptotic approximation algorithm and provides a polynomial-time $(α+ \varepsilon)$-competitive algorithm for online bin packing under the i.i.d. model, where $\varepsilon$>0 is a small constant. Using the AFPTAS for offline bin packing, we thus provide a linear time $(1+\varepsilon)$-competitive algorithm for online bin packing under i.i.d. model, thus settling the problem. We then study the random-order model, where an adversary specifies the items, but the order of arrival of items is drawn uniformly at random from the set of all permutations of the items. Kenyon's seminal result [SODA'96] showed that the Best-Fit algorithm has a competitive ratio of at most 3/2 in the random-order model, and conjectured the ratio to be around 1.15. However, it has been a long-standing open problem to break the barrier of 3/2 even for special cases. Recently, Albers et al. [Algorithmica'21] showed an improvement to 5/4 competitive ratio in the special case when all the item sizes are greater than 1/3. For this special case, we settle the analysis by showing that Best-Fit has a competitive ratio of 1. We make further progress by breaking the barrier of 3/2 for the 3-Partition problem, a notoriously hard special case of bin packing, where all item sizes lie in (1/4,1/2].

Figures (1)

• Figure 1: The division of input into super-stages to get rid of the assumption on the knowledge of $n$. The $(j+1)^{\mathrm{th}}$ super-stage is denoted by $\Gamma_j$. Super-stage $\Gamma_0$ contains $n_0=1/\delta^3$ items, $\Gamma_0\cup \Gamma_1$ contains $(1+\mu)n_0$ items, $\Gamma_0\cup \Gamma_1\cup \Gamma_2$ contains $(1+\mu)^2n_0$ items and so on. The last super-stage may not be full, but since it is very small in size compared to the entire input, it doesn't affect the performance of the algorithm.

Theorems & Definitions (45)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Upright matching with i.i.d. coordinates fischer_thesis
• Lemma 2.2: Upright matching with randomly permuted coordinates fischer_thesis
• Lemma 3.1: Blueprint Packing Lemma
• Lemma 3.2: Bernstein's Inequality
• Lemma 3.3
• proof : Proof Sketch