Bin Packing with Discrete Item Sizes, Part I: Perfect Packing Theorems and the 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 Ufj; kg, 1 < j k; where each item size in f1=k; 2=k; : : : ; j=kg has probability 1=j of being chosen. Note that for xed j; k as m ! 1 the discrete distributions Ufmj; mkg 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 diier 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 Ufj; kg, whereas no on-line algorithm has even o(n 1=2) expected waste under U(0; u] for any 0 < u 1. Our Ufj; kg 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 (n), (n 1=2), or O(1), depending on whether certain \perfect" packings exist. The Perfect Packing Theorem needed for the Ufj; kg distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper.