  We consider two well-known natural variants of bin packing, and show that
these packing problems admit asymptotic fully polynomial time approximation
schemes (AFPTAS). In bin packing problems, a set of one-dimensional items of
size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins).
It has been known for a while that the most basic problem admits an AFPTAS. In
this paper, we develop methods that allow to extend this result to other
variants of bin packing. Specifically, the problems which we study in this
paper, for which we design asymptotic fully polynomial time approximation
schemes, are the following. The first problem is "Bin packing with cardinality
constraints", where a parameter k is given, such that a bin may contain up to k
items. The goal is to minimize the number of bins used. The second problem is
"Bin packing with rejection", where every item has a rejection penalty
associated with it. An item needs to be either packed to a bin or rejected, and
the goal is to minimize the number of used bins plus the total rejection
penalty of unpacked items. This resolves the complexity of two important
variants of the bin packing problem. Our approximation schemes use a novel
method for packing the small items. This new method is the core of the improved
running times of our schemes over the running times of the previous results,
which are only asymptotic polynomial time approximation schemes (APTAS).
