A branch-and-bound algorithm for the two-dimensional vector packing problem
Computers and Operations Research
There is no asymptotic PTAS for two-dimensional vector packing
Information Processing Letters
On multi-dimensional packing problems
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Analysis of Several Task-Scheduling Algorithms for a Model of Multiprogramming Computer Systems
Journal of the ACM (JACM)
Lower bounds and algorithms for the 2-dimensional vector packing problem
Discrete Applied Mathematics
Operations Research Letters
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In this paper we deal with the d-dimensional vector packing problem, which is a generalization of the classical bin packing problem in which each item has d distinct weights and each bin has d corresponding capacities. We address the case in which the vectors of weights associated with the items are totally ordered, i.e., given any two weight vectors ai, aj, either ai is componentwise not smaller than aj or aj is componentwise not smaller than ai, and construct an asymptotic polynomial-time approximation scheme for this case. As a corollary, we also obtain such a scheme for the bin packing problem with cardinality constraint, whose existence was an open question to the best of our knowledge. We also extend the result to instances with constant Dilworth number, i.e. instances where the set of items can be partitioned into a constant number of totally ordered subsets. We use ideas from classical and recent approximation schemes for related problems, as well as a nontrivial procedure to round an LP solution associated with the packing of the small items.