We present an qpproximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classical NP-hard cutting-stock problem. The algorithm find,!; a packing of n rectangles whose total height is within a ,factor of (1 + E ) of optimal, and has running time polynomial both in n and in 1 / E . It is based on a reduction to fractional bin-packing, and ...

We present a new heuristic for 2D-packing of rectangles inside a rectangular area where the aesthetics of the resulting packing is amenable to generating large collages of photographs or images. The heuristic works by maintaining a sorted collection of vertical segments covering the area to be packed. The segments define the leftmost boundaries of rectangular and possibly overlapping areas that...

In this work we describe a message packing and unpacking method for homomorphic ciphertexts. Messages are packed into the coefficients of plaintext polynomials. We propose an unpacking procedure which allows to obtain a ciphertext for each packed message. The packing and unpacking of ciphertexts represents a solution for reducing the transmission bottleneck in cloud based applications, in parti...

An induced packing of odd cycles in a graph is a packing such that there is no edge in the graph between any two odd cycles in the packing. We prove that the problem is solvable in time O(n ) when the input graph is planar. We also show that deciding if a graph has an induced packing of two odd cycles is NP-complete.

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be ro...

In order to verify and test the performance of new packing algorithms relative to existing algorithms, test problems are needed. The scope of published test instances for 2D rectangular and irregular packing has been fairly limited in terms of object size, number and dimension of items. A systematic investigation into the performance of new algorithms requires the use of a range of test problem...

injection molding is the most widely used process for producing plastic parts.in this process, an important issue is to predict and optimize the warpage before manufacturing. in this paper, a simulation with moldflow software is proposed to optimize the injection molding process. the optimization process aims at minimizing the warpage of the critical injection molding part in which process para...

A set of rectangles S is said to be grid packed if there exists a rectangular grid (not necessarily regular) such that every rectangle lies in the grid and there is at most one rectangle of S in each cell. The area of a grid packing is the area of a minimal bounding box that contains all the rectangles in the grid packing. We present an approximation algorithm that given a set S of rectangles a...

A set of rectangles S is said to be grid packed if there exists a rectangular grid (not necessarily regular) such that every rectangle lies in the grid and there is at most one rectangle of S in each cell. The area of a grid packing is the area of a minimal bounding box that contains all the rectangles in the grid packing. We present an approximation algorithm that given a set S of rectangles a...

We study the well-known two-dimensional strip packing problem. Given is a set of rectangular axis-parallel items and a strip of width W with infinite height. The objective is to find a packing of these items into the strip, which minimizes the packing height. Lately, it has been shown that the lower bound of 3/2 of the absolute approximation ratio can be beaten when we allow a pseudo-polynomial...