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 unitsized 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 metaalgorithm that takes an offline αasymptotic proximation algorithm and provides a polynomialtime (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 randomorder 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 BestFit algorithm has a competitive ratio of at most 3/2 in the randomorder model, and conjectured the ratio to be ≈ 1.15. However, it has been a longstanding 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 BestFit has a competitive ratio of 1. We also make further progress by breaking the barrier of 3/2 for the 3Partition problem, a notoriously hard special case of bin packing, where all item sizes lie in (1/4,1/2].
BibTeX  Entry
@InProceedings{ayyadevara_et_al:LIPIcs.ICALP.2022.12,
author = {Ayyadevara, Nikhil ^* and Dabas, Rajni and Khan, Arindam and Sreenivas, K. V. N.},
title = {{NearOptimal Algorithms for Stochastic Online Bin Packing}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {12:112:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772358},
ISSN = {18688969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16353},
URN = {urn:nbn:de:0030drops163532},
doi = {10.4230/LIPIcs.ICALP.2022.12},
annote = {Keywords: Bin Packing, 3Partition Problem, Online Algorithms, Random Order Arrival, IID model, BestFit Algorithm}
}
Keywords: 

Bin Packing, 3Partition Problem, Online Algorithms, Random Order Arrival, IID model, BestFit Algorithm 
Collection: 

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) 
Issue Date: 

2022 
Date of publication: 

28.06.2022 