Greedy algorithms for packing unequal circles into a rectangular container

In this paper, we study the problem of packing unequal circles into a two-dimensional rectangular container. We solve this problem by proposing two greedy algorithms. The first algorithm, denoted by B1.0, selects the next circle to place according to the maximum-hole degree rule, that is inspired from human activity in packing. The second algorithm, denoted by B1.5, improves B1.0 with a self-look-ahead search strategy. The comparisons with the published methods on several instances taken from the literature show the good performance of our approach.     

The densest packing of equal circles into a parallel strip

What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Tóth a number of years ago. The answer is trivial for \(w \leqslant \sqrt 3 /2\) and, surprisingly, it is not difficult to prove [M2] for \(w = n\sqrt 3 /2\) , wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertész [K] solved the case \(w< \sqrt 2 \) . Here we fill the first gap, i.e., the maximal density is determined for \(\sqrt 3 /2< w \leqslant \sqrt 3 \) .     

Quasi-human seniority-order algorithm for unequal circles packing

In the existing methods for solving unequal circles packing problems, the initial configuration is given arbitrarily or randomly, but the impact of different initial configurations for existing packing algorithm to the speed of existing packing algorithm solving unequal circles packing problems is very large. The quasi-human seniority-order algorithm proposed in this paper can generate a better initial configuration for existing packing algorithm to accelerate the speed of existing packing algorithm solving unequal circles packing problems. In experiments, the quasi-human seniority-order algorithm is applied to generate better initial configurations for quasi-physical elasticity methods to solve the unequal circles packing problems, and the experimental results show that the proposed quasi-human seniority-order algorithm can greatly improve the speed of solving the problem.     

Packing unequal circles into a strip of minimal length with a jump algorithm

The paper considers a problem of packing unequal circles into a rectangular strip with fixed width and minimal length. We develop the idea of increasing the dimension of the solution space by assuming radii of circles to be variables. A mathematical model of the problem is constructed and its characteristics are investigated. Taking into account the characteristics we offer a solution strategy of the problem including a number of non-linear programming subproblems of packing circles of variable radii. The solution strategy involves special ways of construction of starting points, calculation of local minima, jump from one local extremum to another, decrease of the problem dimension and rearrangement of pairs of circles. For calculating local extrema an interior point optimizer together with the concept of active inequalities are used. We compare 146 numerical benchmark examples and give seven new ones for 125, 150, 175, 225, 250, 275 and 300 circles.     

On packing squares into a rectangle

We prove that every set of squares with total area 1 can be packed into a rectangle of area at most 2867/2048=1.399… . This improves on the previous best bound of 1.53. Also, our proof yields a linear time algorithm for finding such a packing.     

Exact and approximation algorithms for a soft rectangle packing problem

We consider the problem of finding an arrangement of rectangles with given areas that minimizes the total length of all inner and outer border lines. We present a polynomial time approximation algorithm and derive an upper bound estimation on its approximation ratio. Furthermore, we give a formulation of the problem as mixed-integer nonlinear program and show that it can be approximatively reformulated as linear mixed-integer program. On a test set of problem instances, we compare our approximation algorithm with another one from the literature. Using a standard numerical mixed-integer linear solver, we show that adding the solutions from the approximation algorithm as advanced starter helps to reduce the overall solution time for proven global optimality, or gives better primal and dual bounds if a certain time-limit is reached before.     

Solving the problem of packing equal and unequal circles in a circular container

In this paper we propose a Monotonic Basin Hopping approach and its population-based variant Population Basin Hopping to solve the problem of packing equal and unequal circles within a circular container with minimum radius. Extensive computational experiments have been performed both to analyze the problem at hand, and to choose in an appropriate way the parameter values for the proposed methods. Different improvements with respect to the best results reported in the literature have been detected.     

Local search algorithms for the rectangle packing problem with general spatial costs

We propose local search algorithms for the rectangle packing problem to minimize a general spatial cost associated with the locations of rectangles. The problem is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. Each rectangle has a set of modes, where each mode specifies the width and height of the rectangle and its spatial cost function. The spatial costs are general piecewise linear functions which can be non-convex and discontinuous. Various types of packing problems and scheduling problems can be formulated in this form. To represent a solution of this problem, a pair of permutations of n rectangles is used to determine their horizontal and vertical partial orders, respectively. We show that, under the constraint specified by such a pair of permutations, optimal locations of the rectangles can be efficiently determined by using dynamic programming. The search for finding good pairs of permutations is conducted by local search and metaheuristic algorithms. We report computational results on various implementations using different neighborhoods, and compare their performance. We also compare our algorithms with other existing heuristic algorithms for the rectangle packing problem and scheduling problem. These computational results exhibit good prospects of the proposed algorithms. Key words.rectangle packing – sequence pair – general spatial cost – dynamic programming – metaheuristicsMathematics Subject Classification (1991):20E28, 20G40, 20C20     

Probabilistic analysis of a new class of strip packing algorithms

A new class of algorithms for online packing of rectangles into a strip is proposed and studied. It is proved that the expectation of the unfilled area for this class of algorithms is O(N ^2/3) in the standard (for this type of problems) probabilistic model for N random rectangles.     

Improved local search algorithms for the rectangle packing problem with general spatial costs

The rectangle packing problem with general spatial costs is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. This problem is very general, and various types of packing problems and scheduling problems can be formulated in this form. For this problem, we have previously presented local search algorithms using a pair of permutations of rectangles to represent a solution. In this paper, we propose speed-up techniques to evaluate solutions in various neighborhoods. Computational results for the rectangle packing problem and a real-world scheduling problem exhibit good prospects of the proposed techniques.     

Exact algorithms for the two-dimensional strip packing problem with and without rotations

We propose exact algorithms for the two-dimensional strip packing problem (2SP) with and without 90° rotations. We first focus on the perfect packing problem (PP), which is a special case of 2SP, wherein all given rectangles are required to be packed without wasted space, and design branch-and-bound algorithms introducing several branching rules and bounding operations. A combination of these rules yields an algorithm that is especially efficient for feasible instances of PP. We then propose several methods of applying the PP algorithms to 2SP. Our algorithms succeed in efficiently solving benchmark instances of PP with up to 500 rectangles and those of 2SP with up to 200 rectangles. They are often faster than existing exact algorithms specially tailored for problems without rotations.     

Fast algorithms for greedy triangulation

We present the first quadratic-time algorithm for the greedy triangulation of a finite planar point set, and the first linear-time algorithm for the greedy triangulation of a convex polygon.     

On packing of rectangles in a rectangle

It is known that ${\sum }_{i=1}^{\infty }1∕\left(i(i+1)\right)=1$. In 1968, Meir and Moser (1968) asked for finding the smallest $?$ such that all the rectangles of sizes $1∕i\times 1∕(i+1)$, $i\in \{1,2,\dots \}$, can be packed into a square or a rectangle of area $1+?$. First we show that in Paulhus (1997), the key lemma, as a statement, in the proof of the smallest published upper bound of the minimum area is false, then we prove a different new upper bound. We show that $?\le 1.26?10^{?9}$ if the rectangles are packed into a square and $?\le 6.878?10^{?10}$ if the rectangles are packed into a rectangle.     

Packing non-identical circles within a rectangle with open length

Packing non-identical circles inside a rectangle witnesses a wide range of industrial applications. However, the non-convex constraints in this problem make it intractable using exact analytical approaches. Even via heuristic methods, the solution time for industrial-scale instances sometimes is too long to be acceptable. This article aims to challenge the existing methods for the benchmark instances. The most significant contributions of this work are: firstly, we proposed three types of packing positions for selection and used human intelligence to convert an arbitrary circle sequence into a feasible compact layout; secondly, diverse position selection criteria have been tested, and it is found that the criterion commonly used in the literature is not the best; thirdly, the traditional genetic algorithm is adapted with lower crossover rate but higher mutation rate particularly, and a minor-adjustment operator with the purpose of exploring the neighborhood of the current best solutions is introduced.     

Simple bounds and greedy algorithms for decomposing a flow into a minimal set of paths

Given arbitrary source and target nodes s, t and an s–t-flow defined by its flow-values on each arc of a network, we consider the problem of finding a decomposition of this flow with a minimal number of s–t-paths. This problem is issued from the engineering of telecommunications networks for which the task of implementing a routing solution consists in integrating a set of end-to-end paths. We show that this problem is NP-hard in the strong sense and give some properties of an optimal solution. We then propose upper and lower bounds for the number of paths in an optimal solution. Finally we develop two heuristics based on the properties of a special set of solutions called saturating solutions.     

Polynomial algorithms for guillotine cutting of a rectangle into small rectangles of two kinds

In this paper the problem of optimally guillotine cutting a rectangle (A, B  ) into small rectangles of two kinds is considered. Rectangles of the first kind (c,_ai),i∈I$(c\text{,}{a}_i)\text{,}i\in I$ have the same width, and their heights can be various. Rectangles of the second kind (_bj,d),j∈J$(b_j\text{,}d)\text{,}j\in J$ have the same height, and their widths can be various. The number of occurrences of each small rectangle in a cutting pattern is not restricted. Similar problems often appear in the furniture industry. This cutting problem is reduced to the shortest path problem in a special rectangular grid, for which a linear time algorithm is suggested. This approach generalizes the approach of [E. Girlich, A.G. Tarnowski, On polynomial solvability of two multiprocessor scheduling problems, Mathematical Methods of Operations Research 50 (1999) 27–51; A.G. Tarnowski, Advanced polynomial time algorithm for guillotine generalized pallet loading problem, in: The International Scientific Collection: Decision Making Under Conditions of Uncertainty (Cutting-Packing Problems), Ufa State Aviation Technical University, 1997, pp. 93–124] and allows us to construct polynomial algorithms for the guillotine cutting problem considered with a fixed number of small rectangles of two kinds.     

Efficiently packing unequal disks in a circle

Placing non-overlapping circles in a smallest container is a hard task. In this paper we present our strategy for optimally placing circles in a smallest circle which led us to win an international competition by properly mixing local and global optimization strategies with random search and local moves.     

4-Block heuristic for the rectangle packing problem

In this paper the rectangle packing problem (RPP) is considered. The RPP consists in finding a packing pattern of small rectangles within a larger rectangle such that the area utilization is maximized. We develop new heuristics for the RPP which are based on the G4-heuristic for the pallet loading problem. In addition to the general RPP we take also into account further restrictions which are of practical interest.     

On a posterior evaluation of a simple greedy method for set packing

We consider an approach for ex post evaluation of approximate solutions obtained by a well known simple greedy method for set packing. A performance bound is derived that is a function of the highest average reward per item over subsets as well as the number of allocated subsets and ground items. This a posterior bound can enable much revelation of optimality when the solution is near optimal. One of the advantages of the ex post analysis is that it does not require computing the optimal solution to the LP relaxation. The ex post bound will not be guaranteed to reveal substantial levels of optimality for all problem instances but can be a useful tool that is complementary to other traditional methods for ex post evaluation for the set packing problem.     

Order and static stability into the strip packing problem

This paper investigates the two-dimensional strip packing problem considering the case in which items should be arranged to form a physically stable packing satisfying a predefined item unloading order from the top of the strip. The packing stability analysis is based on conditions for the static equilibrium of rigid bodies, differing from others strategies which are based on area and percentage of support. We consider an integer linear programming model for the strip packing problem with the order constraint, and a cutting plane algorithm to handle stability, leading to a branch-and-cut approach. We also present two heuristics: the first is based on a stack building algorithm; and, the last is a slight modification of the branch-and-cut approach. The computational experiments show that the branch-and-cut model can handle small and medium-sized instances, whereas the heuristics found almost optimal solutions quickly for several instances. With the combination of heuristics and the branch-and-cut algorithm, many instances are solved to near optimality in a few seconds.