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.     

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.     

Expected wasted space of optimal simple rectangle packing

A simple packing of a collection of rectangles contained in [0,1]² is a disjoint subcollection such that each vertical line meets at most one rectangle of the packing. The wasted space of the packing is the surface of the area of the part of [0,1]² not covered by the packing. We prove that for a certain number L, for all N2, the wasted space W_N in an optimal simple packing of N independent uniformly distributed rectangles satisfiesWork partially supported by an N.S.F. grant.Mathematics Subject Classification (2000): 60D05     

Hardness of approximation for orthogonal rectangle packing and covering problems

Bansal and Sviridenko [N. Bansal, M. Sviridenko, New approximability and inapproximability results for 2-dimensional bin packing, in: Proceedings of the 15th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, 2004, pp. 189–196] proved that there is no asymptotic PTAS for 2-dimensional Orthogonal Bin Packing (without rotations), unless P=NP. We show that similar approximation hardness results hold for several 2- and 3-dimensional rectangle packing and covering problems even if rotations by ninety degrees are allowed. Moreover, for some of these problems we provide explicit lower bounds on asymptotic approximation ratio of any polynomial time approximation algorithm. Our hardness results apply to the most studied case of 2-dimensional problems with unit square bins, and for 3-dimensional strip packing and covering problems with a strip of unit square base.     

Optimal multicast route packing

This paper considers the hop-constrained multicast route packing problem with a bandwidth reservation to build QoS-guaranteed multicast routing trees with a minimum installation cost. Given a set of multicast sessions, each of which has a hop limit constraint and a bandwidth requirement, the problem is to determine the set of multicast routing trees in an arc-capacitated network with the objective of minimizing the cost. For the problem, we propose a branch-and-cut-and-price algorithm, which can be viewed as a branch-and-bound method incorporating both the strong cutting plane algorithm and the column generation method. We implemented and tested the proposed algorithm on randomly generated problem instances with sizes up to 30 nodes, 570 arcs, and 10 multicast sessions. The test results show that the algorithm can obtain the optimal solution to practically sized problem instances within a reasonable time limit in most cases.     

A reduction approach for solving the rectangle packing area minimization problem

In the rectangle packing area minimization problem (RPAMP) we are given a set of rectangles with known dimensions. We have to determine an arrangement of all rectangles, without overlapping, inside an enveloping rectangle of minimum area. The paper presents a generic approach for solving the RPAMP that is based on two algorithms, one for the 2D Knapsack Problem (KP), and the other for the 2D Strip Packing Problem (SPP). In this way, solving an instance of the RPAMP is reduced to solving multiple SPP and KP instances. A fast constructive heuristic is used as SPP algorithm while the KP algorithm is instantiated by a tree search method and a genetic algorithm alternatively. All these SPP and KP methods have been published previously. Finally, the best variants of the resulting RPAMP heuristics are combined within one procedure. The guillotine cutting condition is always observed as an additional constraint. The approach was tested on 15 well-known RPAMP instances (above all MCNC and GSRC instances) and new best solutions were obtained for 10 instances. The computational effort remains acceptable. Moreover, 24 new benchmark instances are introduced and promising results are reported.     

Parallel greedy algorithms for packing unequal circles into a strip or a rectangle

Given a finite set of circles of different sizes we study the strip packing problem (SPP) as well as the Knapsack Problem (KP). The SPP asks for a placement of all circles within a rectangular strip of fixed width so that the variable length of the strip is minimized. The KP requires packing of a subset of the circles in a given rectangle so that the wasted area is minimized. To solve these problems some greedy algorithms were developed which enhance the algorithms proposed by Huang et al. (J Oper Res Soc 56:539–548, 2005). Furthermore, the new greedy algorithms were parallelized using a master slave approach. The resulting parallel methods were tested using the instances introduced by Stoyan and Yaskov (Eur J Oper Res 156:590–600, 2004). Additionally, two sets of 128 instances each for the SPP and for the KP were generated and results for these new instances are also reported.     

An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50?000 instances). It is also effective for solving the instances of problem set Cover III (almost 100?000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.     

Computing rectangle enclosures

The rectangle enclosure problem is the problem of determining the subset of n iso-oriented planar rectangles that enclose a query rectangle Q. In this paper, we use a three layered data structure which is a combination of Range and Priority search trees and answers both the static and dynamic cases of the problem. Both the cases use O(n> log² n) space. For the static case, the query time is O(log² n log log n + K). The dynamic case is supported in O(log³ n + K) query time using O(log³ n) amortized time per update. K denotes the size of the answer. For the d-dimensional space the results are analogous. The query time is O(log^2d-2 n log log n + K) for the static case and O(log^2d-1 n + K) for the dynamic case. The space used is O(n> log^2d-2 n) and the amortized time for an update is O(log^2d-1 n). The existing bounds given for a class of problems which includes the present one, are O(log^2d n + K) query time, O(log^2d n) time for an insertion and O(log^2d-1 n) time for a deletion.     

Rectangling a rectangle

We show that the following are equivalent: (i) A rectangle of eccentricityv can be tiled using rectangles of eccentricityu. (ii) There is a rational function with rational coefficients,Q(z), such thatv =Q(u) andQ maps each of the half-planes {z | Re(z) < 0} and {z | Re(z) > 0 into itself, (iii) There is an odd rational function with rational coefficients,Q(z), such thatv = Q(u) and all roots ofv = Q(z) have a positive real part. All rectangles in this article have sides parallel to the coordinate axes and all tilings are finite. We letR(x, y) denote a rectangle with basex and heighty. In 1903 Dehn [1 ] proved his famous result thatR(x, y) can be tiled by squares if and only ify/x is a rational number. Dehn actually proved the following result. (See [4] for a generalization to tilings using triangles.) The first and third authors were partially supported by NSF.     

Parabolic rectangle packings

The contacts graph (or nerve) of a packing is a combinatorial graph which describes the combinatorics of the packing. Let G be the 1-skeleton of a triangulation of an open disk and let P be a rectangle packing with contact graph G. In this paper a topological criterion for deciding whether G is an α-EL parabolic graph is given. Our result shows the internal relation between the topological property of the packing P and the combinatorial property of the contacts graph G of P.     

The rectangle intersection problem revisited

We take another look at the problem of intersecting rectangles with parallel sides. For this we derive a one-pass, time optimal algorithm which is easy to program, generalizes tod dimensions well, and illustrates a basic duality in its data structures and approach.The work of the first author was supported by the DAAD (Deutscher Akademischer Austauschdienst) and carried out while visiting McMaster University as a postdoctoral fellow. The work of the second author was supported by a Natural Sciences and Engineering Research Council of Canada Grant No. A-7700.     

Search game in a rectangle

We consider a search game in which the searcher (player S) moves along a continuous trajectory in a rectangleQ. The velocity vectogram of player S is a rhombus-type set. In this paper, we construct the strategies of both players which make it possible to find the asymptotic value of the game in the case of small discovery radius.The author would like to thank the referee for considerable simplification of the proof of Theorem 4.1.     

On the maximum empty rectangle problem

Given a rectangle A and a set S of n points in A, we consider the problem, called the maximum empty rectangle problem, of finding a maximum area rectangle that is fully contained in A and does not contain any point of S in its interior. An O(n²) time algorithm is presented. Furthermore, it is shown that if the points of S are drawn randomly and independently from A, the problem can be solved in O(n(log n)²) expected time.     

An indefinite Laplacian on a rectangle

In this note, we investigate the nonelliptic differential expression

$$A = - div\operatorname{sgn} \nabla $$

on a rectangular domain Ω in the plane. The seemingly simple problem of associating a self-adjoint operator with the differential expression A in L²(Ω) is solved here. Such indefinite Laplacians arise in mathematical models of metamaterials characterized by negative electric permittivity and/or negative magnetic permeability.

     

Compositions inside a rectangle and unimodality

Let c ^k,l (n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k×l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the sequence c ^k,l (0),c ^k,l (1),…,c ^k,l (kl) is unimodal. The problem of giving a combinatorial proof of this fact is discussed, but is still open.     

Enumeration of polyominoes inscribed in a rectangle

We develop a number of formulas and generating functions for the enumeration of general polyominoes inscribed in a rectangle of given size according to their area. These formulae are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one.