**E. Asgeirsson,
U. Ayesta,
E. Coffman,
J. Etra,
P. Momcilovic,
D. Phillips,
V. Vokshoori,
Z. Wang,
and J. Wolfe**

An optimal algorithm for the classical bin packing problem partitions (packs) a given set of items with sizes at most 1 into a smallest number of unit-capacity bins such that the sum of the sizes of the items in each bin is at most 1. Approximation algorithms for this NP-hard problem are called {\em on-line} if the items are packed sequentially into bins with the bin receiving a given item being independent of the number and sizes of all items as yet unpacked. {\em Off-line} algorithms plan packings assuming full (advance) knowledge of all item sizes. The {\em closed} on-line algorithms are intermediate: item sizes are not known in advance but the number $n$ of items is. The uniform model, where the $n$ item sizes are independent uniform random draws from [0,1], commands special attention in the average-case analysis of bin packing algorithms. In this model, the expected wasted space produced by an optimal off-line algorithm is $\Theta(\sqrt{n})$, while that produced by an optimal on-line algorithm is $\Theta(\sqrt{n \log n})$. Surprisingly, an optimal closed on-line algorithm also wastes only $\Theta(\sqrt{n})$ space on the average. A proof of this last result is the principal contribution of this paper. However, we also identify a class of optimal closed algorithms, extend the main result to other probability models, and give an estimate of the hidden constant factor.

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author = {E. Asgeirsson and U. Ayesta and E. Coffman and J. Etra and P. Mom\v{c}ilovi{\`c} and D. Phillips and V. Vokshoori and Z. Wang and J. Wolfe},

title = {Closed On-Line Bin Packing},

journal = {Acta Cybernetica},

year = {2002},

volume = {15},

pages = {361--367},

number = {3},

abstract = {An optimal algorithm for the classical bin packing problem partitions (packs) a given set of items with sizes at most 1 into a smallest number of unit-capacity bins such that the sum of the sizes of the items in each bin is at most 1. Approximation algorithms for this NP-hard problem are called {\em on-line} if the items are packed sequentially into bins with the bin receiving a given item being independent of the number and sizes of all items as yet unpacked. {\em Off-line} algorithms plan packings assuming full (advance) knowledge of all item sizes. The {\em closed} on-line algorithms are intermediate: item sizes are not known in advance but the number $n$ of items is. The uniform model, where the $n$ item sizes are independent uniform random draws from [0,1], commands special attention in the average-case analysis of bin packing algorithms. In this model, the expected wasted space produced by an optimal off-line algorithm is $\Theta(\sqrt{n})$, while that produced by an optimal on-line algorithm is $\Theta(\sqrt{n \log n})$. Surprisingly, an optimal closed on-line algorithm also wastes only $\Theta(\sqrt{n})$ space on the average. A proof of this last result is the principal contribution of this paper. However, we also identify a class of optimal closed algorithms, extend the main result to other probability models, and give an estimate of the hidden constant factor.}

}