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.
In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be $\mathsf{NP}$-hard. In this paper, we present new sufficient conditions for packing circles into square and triangular containers, using only the sum of the circles' areas: For square containers, it is possible to pack any set of circles with a combined area of up to approximately 53.90% of the square's area. And when the container is a right or obtuse triangle, any set of circles whose combined area does not exceed the triangle's…
This paper addresses the equal circle packing problem, and proposes an efficient Quasi-physical Quasi-human (QPQH) algorithm. QPQH is based on a modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm which we call the local BFGS and a new basin hopping strategy based on a Chinese idiom: alternate tension with relaxation. Starting from a random initial layout, we apply the local BFGS algorithm to reach a local minimum layout. The local BFGS algorithm fully utilizes the neighborhood information of each circle to considerably speed up the running time of the gradient descent process, and the …
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $δ=\frac{8}{5π}\approx 0.509$. This implies that any set of (not necessarily equal) squares of total area $A \leq \frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $\varepsilon>0$ there are sets of squares of total area $\frac{8}{5}+\varepsilon$ that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a cir…
In this work we propose a heuristic algorithm for the layout optimization for disks installed in a rotating circular container. This is a unequal circle packing problem with additional balance constraints. It proved to be an NP-hard problem, which justifies heuristics methods for its resolution in larger instances. The main feature of our heuristic is based on the selection of the next circle to be placed inside the container according to the position of the system's center of mass. Our approach has been tested on a series of instances up to 55 circles and compared with the literature. Computa…
We address the NP-hard problem of finding a non-overlapping dense packing pattern for n Unequal Circle items in a two-dimensional Square Container (PUC-SC) such that the size of the container is minimized. Based on our previous work on an Action Space based Global Optimization (ASGO) that approximates each circle item as a square item to efficiently find the large unoccupied spaces, we propose an optimization algorithm based on the Partitioned Action Space and Partitioned Circle Items (PAS-PCI). The PAS is to partition the narrow action space on the long side to find two equal action spaces to…
This paper proves a bottom-left placement theorem for the rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given rectangles into a rectangular container without overlapping, then we can achieve a feasible packing by successively placing a rectangle onto a bottom-left corner in the container. This theorem shows that even for the real-parameter rectangle packing problem, we can solve it after finite times of bottom-left placement actions. Based on this theorem, we might develop efficient heuristic algorithms for solving the rectangle packing problem.
In this paper, we show an improved bound and new algorithm for the online square-into-square packing problem. This two-dimensional packing problem involves packing an online sequence of squares into a unit square container without any two squares overlapping. The goal is to find the largest area $α$ such that any set of squares with total area $α$ can be packed. We show an algorithm that can pack any set of squares with total area $α\leq 3/8$ into a unit square in an online setting, improving the previous bound of $11/32$.
China Three Gorges University（三峡大学）；Huazhong University of Science & Technology（华中科技大学）；Manchester Metropolitan University（曼彻斯特城市大学）
In this paper we present a new algorithm for a layout optimization problem: this concerns the placement of weighted polygons inside a circular container, the two objectives being to minimize imbalance of mass and to minimize the radius of the container. This problem carries real practical significance in industrial applications (such as the design of satellites), as well as being of significant theoretical interest. Previous work has dealt with circular or rectangular objects, but here we deal with the more realistic case where objects may be represented as polygons and the polygons are allowe…
We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity. We provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings. Using this framework we get a flotilla of results on this problem (and also on its dual, where one wants to pick a maximum weight subset of the points when the regions have capacities). For example, for the …
We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is (2 + ε) [Jansen and Zhang, SODA 2004]. In this paper, we break the 2 approximation barrier, achieving a polynomial-time (17/9 + ε) < 1.89 approximation, which imp…
Cutting and packing problems arise in a large variety of industrial applications, where there is a need to cut pieces from a large object, or placing them inside a containers, without overlap. When the pieces or the containers have irregular outline, the problem is classified as a Nesting problem. The geometrical challenges of the Nesting problem are addressed by focusing on the geometric aspect of the 2D pieces and containers involved. The challenges of the geometrical component are mainly derived from the complexity of the pieces, due to high number of vertices, which is common when dealing …
We explore approximation algorithms for the $d$-dimensional geometric bin packing problem ($d$BP). Caprara (MOR 2008) gave a harmonic-based algorithm for $d$BP having an asymptotic approximation ratio (AAR) of $T_{\infty}^{d-1}$ (where $T_{\infty} \approx 1.691$). However, their algorithm doesn't allow items to be rotated. This is in contrast to some common applications of $d$BP, like packing boxes into shipping containers. We give approximation algorithms for $d$BP when items can be orthogonally rotated about all or a subset of axes. We first give a fast and simple harmonic-based algorithm ha…
We consider the irregular strip packing problem of rasterized shapes, where a given set of pieces of irregular shapes represented in pixels should be placed into a rectangular container without overlap. The rasterized shapes provide simple procedures of the intersection test without any exceptional handling due to geometric issues, while they often require much memory and computational effort in high-resolution. To reduce the complexity of rasterized shapes, we propose a pair of scanlines representation called the double scanline representation that merges consecutive pixels in each row and co…
The three-dimensional bin packing problem (3D-BPP) plays an important role in city logistics and manufacturing environments, due to its direct relevance to operational cost. Most existing literature have investigated the conventional 3D-BPP, in which the shape of items are typically considered as regular shapes, e.g., rectangular-shaped rigid boxes or cylindrical-shaped containers. However, 3D-BPP for non-rectangular shaped items are quite common in varies delivery schemes, especially in fresh food delivery, and few published studies focusing on these issues. In this paper, we address a novel …
A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. We give two polynomial time algorithms, one for deciding if $P$ can be tiled with $k\times k$ squares for any fixed $k$ which can be part of the input (that is, deciding if $P$ is the union of a set of non-overlapping $k\times k$ squares) and one for packing $P$ with a maximum number of non-overlapping and axis-parallel $2\times 1$ dominos, allowing rotations by $9…
Packing optimization is a prevalent problem that necessitates robust and efficient algorithms that are also simple to implement. One group of approaches is the raster methods, which rely on approximating the objects with pixelated representations. Although they are versatile in treating irregular geometries, the raster methods received limited attention in solving problems involving rotatable objects, where available studies generally analyze only right-angled rotations. In addition, raster approximation allows the use of unique performance metrics and indirect consideration of constraints, wh…
This paper discusses the problem of placing weighted items in a circular container in two-dimensional space. This problem is of great practical significance in various mechanical engineering domains, such as the design of communication satellites. Two constructive heuristics are proposed, one for packing circular items and the other for packing rectangular items. These work by first optimizing object placement order, and then optimizing object positioning. Based on these heuristics, an ant colony optimization (ACO) algorithm is described to search first for optimal positioning order, and then …
Huazhong University of Science and Technology（华中科技大学）；Université de Picardie Jules Verne（皮卡第儒勒-凡尔纳大学）
Stochastic gradient descent (SGD) is a powerful method for large-scale optimization problems in the area of machine learning, especially for a finite-sum formulation with numerous variables. In recent years, mini-batch SGD gains great success and has become a standard technique for training deep neural networks fed with big amount of data. Inspired by its success in deep learning, we apply the idea of SGD with batch selection of samples to a classic optimization problem in decision version. Given $n$ unit circles, the equal circle packing problem (ECPP) asks whether there exist a feasible pack…
