Approximation schemes for packing splittable items with cardinality constraints

We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of items must be packed into as few bins as possible. Items may be split, but each bin may contain at most k (parts of) items, where k is some fixed constant. Complicating the problem further is the fact that items may be larger than 1, which is the size of a bin. We close this problem by providing a polynomial-time approximation scheme for it. We first present a scheme for the case k∈=∈2 and then for the general case of constant k. Additionally, we present dual approximation schemes for k∈=∈2 and constant k. Thus we show that for any ε>∈0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1∈+∈ε.