Bin covering with cardinality constraints

Bin covering is the dual problem of bin packing. In this problem, items of size at most 1 are to be partitioned into sets (bins) so as to maximize the number of sets whose total sum is at least 1. Such a bin is called covered. In the problem with cardinality constraints, a parameter k, also called the cardinality constraint, is given. This parameter indicates a lower bound on the number of items that a covered bin must contain in addition to the condition on the total size. Similarly to packing problems, covering problems are typically studied with respect to the asymptotic performance of the algorithms. It is known that a simple greedy algorithm achieves an asymptotic competitive ratio of 2 for the standard online bin covering problem, and no algorithm can have a smaller asymptotic competitive ratio. The standard offline bin covering problem is known to admit an AFPTAS. Our main result is an AFPTAS for bin covering with cardinality constraints. We further study online bin covering with cardinality constraints, and show that this problem is strictly harder than the standard problem (for any k>4) by providing a lower bound of 52-2k on the asymptotic competitive ratio of any online algorithm. This lower bound holds even if the items are presented sorted according to size, in a non-increasing order. We design an algorithm with an asymptotic competitive ratio which can be made arbitrarily close to 3-2k, for any cardinality constraint k. We show that the special case k=2 admits tight bounds of 2 on the asymptotic competitive ratio. Finally, we study a semi-online variant with non-decreasing sizes and show tight bounds of 2 on its asymptotic competitive ratio for any value of k.