Using dual approximation algorithms for scheduling problems theoretical and practical results
Journal of the ACM (JACM)
An efficient approximation scheme for variable-sized bin packing
SIAM Journal on Computing
Approximation algorithms for bin packing: a survey
Approximation algorithms for NP-hard problems
Analysis of Several Task-Scheduling Algorithms for a Model of Multiprogramming Computer Systems
Journal of the ACM (JACM)
On-line Packing and Covering Problems
Developments from a June 1996 seminar on Online algorithms: the state of the art
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
On strip packing With rotations
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Online Bin Packing with Cardinality Constraints
SIAM Journal on Discrete Mathematics
A fast asymptotic approximation scheme for bin packing with rejection
Theoretical Computer Science
An efficient approximation scheme for the one-dimensional bin-packing problem
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Bin packing problems with rejection penalties and their dual problems
Information and Computation
Bin Packing with Rejection Revisited
Algorithmica
Bin packing with general cost structures
Mathematical Programming: Series A and B
Fast asymptotic FPTAS for packing fragmentable items with costs
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
Set covering with ordered replacement: additive and multiplicative gaps
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Robust algorithms for preemptive scheduling
ESA'11 Proceedings of the 19th European conference on Algorithms
The entropy rounding method in approximation algorithms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Bin covering with cardinality constraints
Discrete Applied Mathematics
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We consider two well-known natural variants of bin packing and show that these packing problems admit asymptotic fully polynomial time approximation schemes (AFPTASs). 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 new methods that allow us to extend this result to other variants of bin packing consisting of a family of two-parameter bin packing problems. We demonstrate the use of our methods by designing AFPTASs for the following problems. 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 bins used and 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 approximationschemes.