We present approximation algorithms for the three-dimensional strip packing problem, and the three-dimensional bin packing problem. We consider orthogonal packings where ninety-degree rotations are allowed. The algorithms we show for these problems have asymptotic performance bounds 2.64, and 4.89, respectively. These algorithms are for the more general case in which the bounded dimensions of t...

We introduce a constraint for one-dimensional bin packing. This constraint uses propagation rules incorporating knapsack-based reasoning, as well as a lower bound on the number of bins needed. We show that this constraint can significantly reduce search on bin packing problems. We also demonstrate that when coupled with a standard bin packing search strategy, our constraint can be a competitive...

In the dual bin packing problem, the objective is to assign items of given size to the largest possible number of bins, subject to the constraint that the total size of the items assigned to any bin is at least equal to 1. We carry out a probabilistic analysis of this problem under the assumption that the items are drawn independently from the uniform distribution on [0,1], and reveal the conne...

It is well known that the two simple algorithms for the classic bin packing problem, NF and WF both have an approximation ratio of 2. However, WF seems to be a more reasonable algorithm, since it never opens a new bin if an existing bin can still be used. Using resource augmented analysis, where the output of an approximation algorithm, which can use bins of size b > 1, is compared to an optima...

The state-of-the-art global constraint for bin packing is due to Shaw. We compare two linear continuous relaxations of the bin packing problem, based on the DP-flow and Arc-flow models, with the filtering of the bin packing constraint. Our experiments show that we often obtain significant improvements in runtime. The DP-flow model is a novel formulation of the problem.

Consider a manifold constructed by identifying the boundaries of Euclidean triangles or Euclidean tetrahedra. When these form a closed topological manifold, we call such spaces piecewise flat manifolds (see Definition 1) as in [8]. Such spaces may be considered discrete analogues of Riemannian manifolds, in that their geometry can be described locally by a finite number of parameters, and the s...

Thurston’s circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to the circle. The main proofs of the uniformization use hyperbolic volumes (Andr...

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [Mil], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one ...

In this paper we present approximation algorithms for the two dimensional strip packing problem with unloading constraints. In this problem, we are given a strip S of width 1 and unbounded height, and n items of C different classes, each item ai with height h(ai), width w(ai) and class c(ai). As in the strip packing problem, we have to pack all items minimizing the used height, but now we have ...

Given bins of size B, non-negative values d and ∆, and a list L of items, each item e ∈ L with size se and class ce, we define a shelf as a subset of items packed inside a bin with total items size at most ∆ such that all items in this shelf have the same class. Two subsequent shelves must be separated by a shelf division of size d. The size of a shelf is the total size of its items plus the si...