SIAM Journal on Computing
An efficient approximation scheme for variable-sized bin packing
SIAM Journal on Computing
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Approximation schemes for covering and packing problems in image processing and VLSI
Journal of the ACM (JACM)
Approximation algorithms for bin packing: a survey
Approximation algorithms for NP-hard problems
On multi-dimensional packing problems
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
`` Strong '' NP-Completeness Results: Motivation, Examples, and Implications
Journal of the ACM (JACM)
Better approximation algorithms for bin covering
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
A Near-Optimal Solution to a Two-Dimensional Cutting Stock Problem
Mathematics of Operations Research
Minimizing the makespan on a batch machine with non-identical job sizes: an exact procedure
Computers and Operations Research
Packing 2-Dimensional Bins in Harmony
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes
Mathematics of Operations Research
An efficient approximation scheme for the one-dimensional bin-packing problem
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
An APTAS for Generalized Cost Variable-Sized Bin Packing
SIAM Journal on Computing
PTAS for k-Tour Cover Problem on the Plane for Moderately Large Values of k
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Computers and Operations Research
An asymptotic PTAS for batch scheduling with nonidentical job sizes to minimize makespan
COCOA'07 Proceedings of the 1st international conference on Combinatorial optimization and applications
Bin packing with general cost structures
Mathematical Programming: Series A and B
The entropy rounding method in approximation algorithms
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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We consider the train delivery problem which is a generalization of the bin packing problem and is equivalent to a one dimensional version of the vehicle routing problem with unsplittable demands. The problem is also equivalent to the problem of minimizing the makespan on a single batch machine with non-identical job sizes. The train delivery problem is strongly NP-hard and does not admit an approximation ratio better than 3/2. We design the first approximation schemes for the general problem. We give an asymptotic polynomial time approximation scheme, under a notion of asymptotic that makes sense even though scaling can cause the cost of the optimal solution of any instance to be arbitrarily large. Alternatively, we give a polynomial time approximation scheme for the case where W, an input parameter that corresponds to the bin size or the vehicle capacity, is polynomial in the number of items or demands.