# Near-Optimal Algorithms for Stochastic Online Bin Packing

Ayyadevara, N and Dabas, R and Khan, A and Sreenivas, KVN (2022) Near-Optimal Algorithms for Stochastic Online Bin Packing. In: 49th EATCS International Conference on Automata, Languages, and Programming, ICALP 2022, 4 - 8 July 2022, Paris.

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Official URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.12

## 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 proximation algorithm and provides a polynomial-time (α + ε)-competitive algorithm for online bin packing under the i.i.d. model, where ε > 0 is a small constant. Using the AFPTAS for offline bin packing, we thus provide a linear time (1 + ε)-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 ≈ 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 also 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].

Item Type: Conference Paper Leibniz International Proceedings in Informatics, LIPIcs Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing The copyright for this article belongs to the Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. Polynomial approximation; Stochastic models, 3-partition problem; Best fit algorithm; Bin packing; Bin packing problem; Competitive ratio; IID model; On-line algorithms; Online bin packing; Partition problem; Random order arrival, Stochastic systems Division of Electrical Sciences > Computer Science & Automation 20 Jul 2022 12:00 20 Jul 2022 12:00 https://eprints.iisc.ac.in/id/eprint/74941