Bin Packing with Discrete Item Sizes, Part I: Perfect Packing Theorems and the Average Case Behavior of Optimal Packings

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}, $1 < j \leq k,$ 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 $m\rightarrow\infty$ 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 $0 < u \leq 1$. 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 $\Theta (n)$, $\Theta (n^{1/2} )$, 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.