We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of xed width and minimum height, a classical NP-hard cutting-stock problem. The algorithm nds a packing of n rectangles whose total height is within a factor of (1 +) of optimal (up to an additive term), and has running time polynomial both in n and in 1==. It is based on a r...

The article is devoted to mathematical models for solving cutting and packing (C&P) problems. We review the main tool of our studies – phi-functions. Those are constructed here for 2D and 3D objects (unlike other standard tools, such as No-Fit Polygons, which are restricted to the 2D geometry). We also demonstrate that in many realistic cases the phi-functions can be described by quite simple f...

This report describes a recursive process for generating data sets of rigid rectangles that can be placed into rectangular regions with zero waste. The generation procedure can be modified to guarantee that the aspect and area ratios of the rectangles in the generated data sets satisfy user-specified parameters. This recursive process can thus be employed to create a variety of data sets that c...

The problem of rectangular objects packing into a semi-endless strip is under consideration. Here we propose to solve this problem by linear cutting approximation that may be presented as block structure of packing. Sometimes it becomes necessary to rearrange elements inside linear cutting blocks. The determined method of “reconstruction”(REC) proposed below helps to get solutions close to opti...

In computational geometry, packing problems ask whether a set of rigid pieces can be placed inside a target region such that no two pieces overlap. The triangle packing problem is a packing problem that involves triangular pieces, and it is crucial for algorithm design in many areas, including origami design, cutting industries, and warehousing. Previous works in packing algorithms have conject...

The facet of Knapsack ploytope, i.e. convex hull of 0-1 points satisfying a given linear inequality has been presented in this current paper. Such type of facets plays an important role in set covering set partitioning, matroidal-intersection vertex- packing, generalized assignment and other combinatorial problems. Strong covers for facets of Knapsack ploytope has been developed in the first pa...

We survey the main formulations and solution methods for two-dimensional orthogonal cutting packing problems, where both items bins are rectangles. focus on exact relaxations four problems from literature: finding a with minimum height, into number of bins, maximum value, determining existence feasible packing.

Consider the feasibility problem in higher-dimensional orthogonal packing. Given a set I of d-dimensional rectangles, we need to decide whether a feasible packing in a d-dimensional rectangular container is possible. No item rotation is allowed and item edges are parallel to the coordinate axes. Typically, solution methods employ some bounds to facilitate the decision. Various bounds are known,...