Bin packing with discrete item sizes, part I: Perfect packing theorems andthe average case behavior of optimal packings

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 less than or equal to 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 --> i nfinity the discrete distributions U{mj, mk} approach the continuous distri bution U(0, j/k], where the item sizes are chosen uniformly from the interv al (0,j/k]. We show that average-case behavior can differ substantially bet ween 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 spa ce under U{j, k}, whereas no on-line algorithm has even o(n(1/2)) expected waste under U(0, u] for any 0 < u. I 1. Our U{j, k} result is an applicatio n 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 -(n), -(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 in dependent combinatorial interest, and its proof is a cornerstone of the pap er.