Online Bin Packing with Cardinality Constraints

Abstract

We consider a one dimensional storage system where each container can store a bounded amount of capacity as well as a bounded number of items k ≥ 2. This defines the (standard) bin packing problem with cardinality constraints which is an important version of bin packing, introduced by Krause, Shen and Schwetman already in 1975. Following previous work on the unbounded space online problem, we establish the exact best competitive ratio for bounded space online algorithms for every value of k. This competitive ratio is a strictly increasing function of k which tends to ${\it \Pi}_{\infty}+1\approx 2.69103$ for large k. Lee and Lee showed in 1985 that the best possible competitive ratio for online bounded space algorithms for the classical bin packing problem is the sum of a series, and tends to ${\it \Pi}_{\rm \infty}$ as the allowed space (number of open bins) tends to infinity. We further design optimal online bounded space algorithms for variable sized bin packing, where each allowed bin size may have a distinct cardinality constraint, and for the resource augmentation model. All algorithms achieve the exact best possible competitive ratio possible for the given problem, and use constant numbers of open bins. Finally, we introduce unbounded space online algorithms with smaller competitive ratios than the previously known best algorithms for small values of k, for the standard cardinality constrained problem. These are the first algorithms with competitive ratio below 2 for k = 4,5,6.