In the bin packing problem, we are asked to place given items, each being of size between zero and one, into bins of capacity one. The goal is to minimize the number of bins that contain at least one item. An online algorithm for the bin packing problem decides where to place each item one by one when it arrives. The asymptotic approximation ratio of the bin packing problem is defined as the performance of an optimal online algorithm for the problem. That value indicates the intrinsic hardness of the bin packing problem. In this paper we study the bin packing problem in which every item is of either size α or size β (≤ α). While the asymptotic approximation ratio for $alpha > rac{1}{2}$ was already identified, that for $alpha leq rac{1}{2}$ is only partially known. This paper is the first to give a lower bound on the asymptotic approximation ratio for any $alpha leq rac{1}{2}$, by formulating linear optimization problems. Furthermore, we derive another lower bound in a closed form by constructing dual feasible solutions.
Hiroshi FUJIWARA
Shinshu University
Ken ENDO
Shinshu University
Hiroaki YAMAMOTO
Shinshu University
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Hiroshi FUJIWARA, Ken ENDO, Hiroaki YAMAMOTO, "Analysis of Lower Bounds for Online Bin Packing with Two Item Sizes" in IEICE TRANSACTIONS on Fundamentals,
vol. E104-A, no. 9, pp. 1127-1133, September 2021, doi: 10.1587/transfun.2020DMP0007.
Abstract: In the bin packing problem, we are asked to place given items, each being of size between zero and one, into bins of capacity one. The goal is to minimize the number of bins that contain at least one item. An online algorithm for the bin packing problem decides where to place each item one by one when it arrives. The asymptotic approximation ratio of the bin packing problem is defined as the performance of an optimal online algorithm for the problem. That value indicates the intrinsic hardness of the bin packing problem. In this paper we study the bin packing problem in which every item is of either size α or size β (≤ α). While the asymptotic approximation ratio for $alpha > rac{1}{2}$ was already identified, that for $alpha leq rac{1}{2}$ is only partially known. This paper is the first to give a lower bound on the asymptotic approximation ratio for any $alpha leq rac{1}{2}$, by formulating linear optimization problems. Furthermore, we derive another lower bound in a closed form by constructing dual feasible solutions.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2020DMP0007/_p
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@ARTICLE{e104-a_9_1127,
author={Hiroshi FUJIWARA, Ken ENDO, Hiroaki YAMAMOTO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Analysis of Lower Bounds for Online Bin Packing with Two Item Sizes},
year={2021},
volume={E104-A},
number={9},
pages={1127-1133},
abstract={In the bin packing problem, we are asked to place given items, each being of size between zero and one, into bins of capacity one. The goal is to minimize the number of bins that contain at least one item. An online algorithm for the bin packing problem decides where to place each item one by one when it arrives. The asymptotic approximation ratio of the bin packing problem is defined as the performance of an optimal online algorithm for the problem. That value indicates the intrinsic hardness of the bin packing problem. In this paper we study the bin packing problem in which every item is of either size α or size β (≤ α). While the asymptotic approximation ratio for $alpha > rac{1}{2}$ was already identified, that for $alpha leq rac{1}{2}$ is only partially known. This paper is the first to give a lower bound on the asymptotic approximation ratio for any $alpha leq rac{1}{2}$, by formulating linear optimization problems. Furthermore, we derive another lower bound in a closed form by constructing dual feasible solutions.},
keywords={},
doi={10.1587/transfun.2020DMP0007},
ISSN={1745-1337},
month={September},}
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TY - JOUR
TI - Analysis of Lower Bounds for Online Bin Packing with Two Item Sizes
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1127
EP - 1133
AU - Hiroshi FUJIWARA
AU - Ken ENDO
AU - Hiroaki YAMAMOTO
PY - 2021
DO - 10.1587/transfun.2020DMP0007
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E104-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2021
AB - In the bin packing problem, we are asked to place given items, each being of size between zero and one, into bins of capacity one. The goal is to minimize the number of bins that contain at least one item. An online algorithm for the bin packing problem decides where to place each item one by one when it arrives. The asymptotic approximation ratio of the bin packing problem is defined as the performance of an optimal online algorithm for the problem. That value indicates the intrinsic hardness of the bin packing problem. In this paper we study the bin packing problem in which every item is of either size α or size β (≤ α). While the asymptotic approximation ratio for $alpha > rac{1}{2}$ was already identified, that for $alpha leq rac{1}{2}$ is only partially known. This paper is the first to give a lower bound on the asymptotic approximation ratio for any $alpha leq rac{1}{2}$, by formulating linear optimization problems. Furthermore, we derive another lower bound in a closed form by constructing dual feasible solutions.
ER -