A simple on-line bin-packing algorithm
Journal of the ACM (JACM)
Online algorithms for a dual version of bin packing
Discrete Applied Mathematics
There is no asymptotic PTAS for two-dimensional vector packing
Information Processing Letters
Using fast matrix multiplication to find basic solutions
Theoretical Computer Science
Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Analysis of Several Task-Scheduling Algorithms for a Model of Multiprogramming Computer Systems
Journal of the ACM (JACM)
Better approximation algorithms for bin covering
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On the online bin packing problem
Journal of the ACM (JACM)
An asymptotic fully polynomial time approximation scheme for bin covering
Theoretical Computer Science
Algorithms for on-line bin-packing problems with cardinality constraints
Discrete Applied Mathematics
Online Bin Packing with Cardinality Constraints
SIAM Journal on Discrete Mathematics
Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems
SIAM Journal on Computing
An efficient approximation scheme for the one-dimensional bin-packing problem
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
AFPTAS Results for Common Variants of Bin Packing: A New Method for Handling the Small Items
SIAM Journal on Optimization
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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 k4) 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.