Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

Abstract

We consider the one-dimensional bin packing problem under the discrete uniform distributions $U\{j,k\}$, $1 \leq j \leq k-1$, in which the bin capacity is $k$ and item sizes are chosen uniformly from the set $\{1,2,\ldots,j\}$. Note that for $0 < u = j/k \leq 1$ this is a discrete version of the previously studied continuous uniform distribution $U(0,u]$, where the bin capacity is 1 and item sizes are chosen uniformly from the interval $(0,u]$. We show that the average-case performance of heuristics can differ substantially between the two types of distributions. In particular, there is an online algorithm that has constant expected wasted space under $U\{j,k\}$ for any $j,k$ with $1 \leq j < k-1$, whereas no online algorithm can have $o(n^{1/2})$ expected waste under $U(0,u]$ for any $0 < u \leq 1$. Our $U\{j,k\}$ result is an application of a general theorem of Courcoubetis and Weber that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of $n$ items must be either $\Theta (n)$, $\Theta (n^{1/2} )$, 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. We also survey other recent results comparing the behavior of heuristics under discrete and continuous uniform distributions.