  We consider the following generalization of the bin packing problem. We are
given a set of items each of which is associated with a rational size in the
interval [0,1], and a monotone non-decreasing non-negative cost function f
defined over the cardinalities of the subsets of items. A feasible solution is
a partition of the set of items into bins subject to the constraint that the
total size of items in every bin is at most 1. Unlike bin packing, the goal
function is to minimize the total cost of the bins where the cost of a bin is
the value of f applied on the cardinality of the subset of items packed into
the bin. We present an APTAS for this strongly NP-hard problem. We also provide
a complete complexity classification of the problem with respect to the choice
of f.
