Approximating Smallest Containers for Packing Three-dimensional Convex Objects

We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NP-hard so that we cannot expect to find exact polynomial time algorithms. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimal volume containers for the objects described

An iterated local search algorithm based on nonlinear programming for the irregular strip packing problem

The irregular strip packing problem is a combinatorial optimization problem that requires to place a given set of two-dimensional polygons within a rectangular container so that no polygon overlaps with other polygons or protrudes from the container, where each polygon is not necessarily convex. The container has a fixed width, while its length can change so that all polygons are placed in it. The objective is to find a layout of the set of polygons that minimizes the length of the container. We propose an algorithm that separates overlapping polygons based on nonlinear programming, and an algorithm that swaps two polygons in a layout so as to find their new positions in the layout with the least overlap. We incorporate these algorithms as components into an iterated local search algorithm for the overlap minimization problem and then develop an algorithm for the irregular strip packing problem using the iterated local search algorithm. Computational comparisons on representative instances disclose that our algorithm is competitive with other existing algorithms. Moreover, our algorithm updates several best known results

Optimal packing of convex polytopes using quasi-phi-functions

We study a packing problem of a given collection of convex polytopes into a rectangular container of minimal volume. Continuous rotations and translations of polytopes are allowed. In addition a given minimal allowable distances between polytopes are taking into account. We employ radical free quasi-phi-functions and adjusted quasi-phi-functions to describe placement constraints. The use of quasi-phi-functions, instead of phi-functions, allows us to simplify non-overlapping, as well as, to describe distance constraints, but there is a price to pay: now the optimization has to be performed over a larger set of parameters, including the extra variables used by our new functions. We provide an exact mathematical model of the problem as a nonlinear programming problem. We also develop an efficient solution algorithm which involves a starting point algorithm, using homothetic trasformations of geometric objects and efficient local optimization procedure, which allows us to runtime and memory). We present here a number of examples to demonstrate the efficiency of our methodology.Рассматривается задача упаковки выпуклых многоранников в прямоугольном контейнере минимального объема. Допускаются непрерывные трансляции и повороты многогранников. Учитываются минимально допустимые расстояния, заданные между многогранниками. Для формализвации ограничений размещения применяются свободные от радикалов квази-phi-функции и псевдонормализованные квази-phi-функции. Использование квази-phi-функций, вместо phi-функций, позволяет упростить вид ограничений непересечения многогранников и описать в аналитическом виде ограничения на минимально допустимые расстояния, заданные между многогранниками. Однако процесс оптимизации требует большего числа параметров, включая дополнительные переменные для квази-phi-функций. Строится математическая модель в виде задачи нелинейного программирования. Предлагается эффективный метод решения, включающий: алгоритм, основанный на гомотетических преобразованиях геометрических объектов для построения допустимых стартовых точек, и процедура локальной оптимизации, которая позволила значительно уменьшить размерность задачи, а также сократить вычислительные ресурсы (время и память). Приводятся результаты численных экспериментов, которые демонстрируют эффективность предложенной методологии решения задачи упаковки выпуклых многогранников.Розглядається задача упаковки опуклих багатогранників у прямокутний контейнер мінімального об’єму. При цьому багатогранники припускають безперервні повороти та трансляції. Крім того, враховуються мінімально припустимі відстані між багатогранниками. Для побудови математичної моделі задачі як задачі нелінійного програмування використовуються вільні від радикалів квазі-phi-функції. Розроблено ефективний алгоритм розв’язання, який дозволяє зменшити розмірність задачі і обчислювальні витрати. Наведено числові приклади

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Packing of concave polyhedra with continuous rotations using nonlinear optimisation

We study the problem of packing a given collection of arbitrary, in general concave, polyhedra into a cuboid of minimal volume. Continuous rotations and translations of polyhedra are allowed. In addition, minimal allowable distances between polyhedra are taken into account. We derive an exact mathematical model using adjusted radical free quasi phi-functions for concave polyhedra to describe non-overlapping and distance constraints. The model is a nonlinear programming formulation. We develop an efficient solution algorithm, which employs a fast starting point algorithm and a new compaction procedure. The procedure reduces our problem to a sequence of nonlinear programming subproblems of considerably smaller dimension and a smaller number of nonlinear inequalities. The benefit of this approach is borne out by the computational results, which include a comparison with previously published instances and new instances