Closed On-Line Bin Packing
E. Asgeirsson, U. Ayesta, E. Coffman, J. Etra, P. Momcilovic, D. Phillips, V. Vokhshoori, Z. Wang, and J. Wolfe
Acta Cybernetica 15 (2002) 361-367.
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 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. Off-line algorithms plan packings assuming full (advance) knowledge of all item sizes. The 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( n 1/2), while that produced by an optimal on-line algorithm is Theta(( n log n )1/2). Surprisingly, an optimal closed on-line algorithm also wastes only Theta( n 1/2) 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.