Bin packing with discrete item sizes, part I: perfect packing theorems and the average case behavior of optimal packings

Document Type

Article

Publication Date

1-1-2000

Abstract

We consider the one-dimensional bin packing problem with unit-capacity bins and item sizes chosen according to the discrete uniform distribution U{j, k}, 1 < j ≤ k, where each item size in {1/k, 2/k, . . . , j/k} has probability 1/j of being chosen. Note that for fixed j, k as m → ∞ the discrete distributions U{mj, mk} approach the continuous distribution U(0, j/k], where the item sizes are chosen uniformly from the interval (0, j/k] We show that average-case behavior can differ substantially between the two types of distributions. In particular, for all j, k with j < k - 1, there exist on-line algorithms that have constant expected wasted space under U{j, k}, whereas no on-line algorithm has even o(n1/2) expected waste under U(0, u] for any 0 < u ≤ 1. Our U{j, k} result is an application of a general theorem of Courcoubetis and Weber [C. Courcoubetis and R.R. Weber, Probab. Engrg. Inform. Sci., 4 (1990), pp. 447-460] that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of n items must be either Θ(n), Θ(n1/2), or O(1), depending on whether certain "perfect" packings exist. The perfect packing theorem needed for the U{j, k} distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper.

Identifier

0010063130 (Scopus)

Publication Title

SIAM Journal on Discrete Mathematics

External Full Text Location

https://doi.org/10.1137/S0895480197325936

ISSN

08954801

First Page

384

Last Page

402

Issue

3

Volume

13

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