By rectangle packing we mean putting a set of rectangles into an enclosing rectangle, without any overlapping. We begin with perfect rectangle packing problems, then prove two continuity properties for parallel rectangle packing problems, and discuss how they might be used to obtain negative results for perfect rectangle packing problems.
We slightly improve the known lower bound on the asymptotic competitive ratio for online bin packing of rectangles. We present a complete proof for the new lower bound, whose value is above 1.91.
It is known that $\sum\limits_{i=1}^{\infty} \frac{1}{i (i+1)} = 1$. In 1968, Meir and Moser asked for finding the smallest $ε$ such that all the rectangles of sizes $1/i \times 1/(i + 1)$ for $i = 1, 2, \ldots$, can be packed into a unit square or a rectangle of area $1 + ε$. In this paper, we show that we can pack the first $1.35\cdot10^{11}$ rectangles into the unit square and give an estimate for $ε$ from this packing.
Given a finite set S in $[0,1]^2$ including the origin, an anchored rectangle packing is a set of non-overlapping rectangles in the unit square where each rectangle has a point of S as its left-bottom corner and contains no point of S in its interior. Allen Freedman conjectured in the 1960's one can always find an anchored rectangle packing with total area at least $1/2$. We verify the conjecture for point configurations whose relative positions belong to certain classes of permutations.
莫泽问，尺寸为1 x 1/2，1/2 x 1/3，1/3 x 1/4，...，总面积等于1的正方形的集合是否可以无重叠地装入单位正方形，边长为1/2，1/3，1/4，.可以无重叠地组合成一个面积为pi ^2/6-1的矩形。人们已经分别对将这些集合包装成边长为1+ π的正方形和面积为π ^2/6-1+ π的矩形进行了计算研究，并且可以考虑这样一个显然较弱的问题：对于每个正数π，这种包装是否可能。在本文中，我们建立了一个一般定理序列的几何包装，这意味着特别是“为每一个ε”的版本，这两个问题实际上是等价的原始平铺问题。
Let $\mathcal{R}$ be a family of axis-parallel rectangles in the plane. The transversal number $τ(\mathcal{R})$ is the minimum number of points needed to pierce all the rectangles. The independence number $ν(\mathcal{R})$ is the maximum number of pairwise disjoint rectangles. Given a positive real number $r$, we say that $\mathcal{R}$ is an r-bounded family if, for any rectangle in $\mathcal{R}$, the aspect ratio of the longer side over the shorter side is at most $r$. Gyárfás and Lehel asked if it is possible to bound the transversal number $τ(\mathcal{R})$ with a linear function of the indep…
It is known that $\sum\limits_{i =1}^\infty {1/ i^2}={π^2/6}$. Meir and Moser asked what is the smallest $ε$ such that all the squares of sides of length $1$, $1/2$, $1/3$, $\ldots$ can be packed into a rectangle of area ${π^2/6}+ε$. A packing into a rectangle of the right area is called perfect packing. Chalcraft packed the squares of sides of length $1$, $2^{-t}$, $3^{-t}$, $\ldots$ and he found perfect packing for $1/2<t\le3/5$. We will show based on an algorithm by Chalcraft that there are perfect packings if $1/2<t\le2/3$. Moreover we show that there is a perfect packing for all $t$ in th…
We survey the main formulations and solution methods for two-dimensional orthogonal cutting and packing problems, where both items and bins are rectangles. We focus on exact methods and relaxations for the four main problems from the literature: finding a packing with minimum height, packing the items into the minimum number of bins, finding a packing of maximum value, and determining the existence of a feasible packing.
We are concerned with the problem of designing large families of subsets over a common labeled ground set that have small pairwise intersections and the property that the maximum discrepancy of the label values within each of the sets is less than or equal to one. Our results, based on transversal designs, factorizations of packings and Latin rectangles, show that by jointly constructing the sets and labeling scheme, one can achieve optimal family sizes for many parameter choices. Probabilistic arguments akin to those used for pseudorandom generators lead to significantly suboptimal results wh…
