Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
The solution of some random NP-hard problems in polynomial expected time
Journal of Algorithms
Approximation algorithms for bin packing: a survey
Approximation algorithms for NP-hard problems
There is no asymptotic PTAS for two-dimensional vector packing
Information Processing Letters
Packing 2-Dimensional Bins in Harmony
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
A probabilistic analysis of multidimensional bin packing problems
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
On Multidimensional Packing Problems
SIAM Journal on Computing
Improved approximation algorithms for multidimensional bin packing problems
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes
Mathematics of Operations Research
Smoothed analysis: an attempt to explain the behavior of algorithms in practice
Communications of the ACM - A View of Parallel Computing
Rectangle packing with one-dimensional resource augmentation
Discrete Optimization
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The multidimensional bin packing and vector bin packing problems are known to not have asymptotic polynomial-time approximation schemes (unless P = NP). Nevertheless, we show that: • Any smoothed (randomly perturbed) instance, and any instance from a class of other distributions, does have a polynomial-time probable approximation scheme. Namely, for any fixed ε 0, we exhibit a linear-time algorithm that finds a (1+ ε)-approximate packing with probability 1 - 2-Ω(n) over the space of random inputs. • There exists an oblivious algorithm that does not know from which distribution inputs come, and still asymptotically does almost as well as the previous algorithms. The oblivious algorithm outputs almost surely a (1 + ε)-approximation for every ε 0. • For vector bin packing, for each considered class of random instances, there exists an algorithm that in expected linear time computes a (1 + ε)-approximation, for any fixed ε 0. To achieve these results we develop a multidimensional version of the one-dimensional rounding technique introduced by Fernadez de la Vega and Lueker. Our results generalize Karp, Luby and Marchetti-Spaccamela's results on approximatibility of random instances of multidimensional bin packing to a much wider class of distributions.