An Algebraic Approach to Rectangle Packing Problems

On the union complexity of families of axis-parallel rectangles with a low packing number

Let R be a family of n axis-parallel rectangles with packing number p-1, meaning that among any p of the rectangles, there are two with a non-empty intersection. We show that the union complexity of R is at most O(n+p^2), and that the (<=k)-level complexity of R is at most O(kn+k^2p^2). Both upper bounds are tight.

A lower bound for online rectangle packing

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.

Corner Occupying Theorem for the Two-dimensional Integral Rectangle Packing Problem

This paper proves a corner occupying theorem for the two-dimensional integral rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given integral rectangles into an integral rectangular container without overlapping, then we can achieve a feasible packing by successively placing an integral rectangle onto a bottom-left corner in the container. Based on this theorem, we might develop efficient heuristic algorithms for solving the integral rectangle packing problem. In fact, as a vague conjecture, this theorem has been implicitly mentioned with different …

Packing Ferrers Shapes

Answering a question of Wilf, we show that if $n$ is sufficiently large, then one cannot cover an $n \times p(n)$ rectangle using each of the $p(n)$ distinct Ferrers shapes of size $n$ exactly once. Moreover, the maximum number of pairwise distinct, non-overlapping Ferrers shapes that can be packed in such a rectangle is only $Θ(p(n)/ \log n).$

Packing $1.35\cdot 10^{11}$ rectangles into a unit square

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.

An Existential Proof of the Conjecture on Packing Anchored Rectangles

Let $P_{n}$ be a set of $n$ points, including the origin, in the unit square $U = [0,1]^2$. We consider the problem of constructing $n$ axis-parallel and mutually disjoint rectangles inside $U$ such that the bottom-left corner of each rectangle coincides with a point in $P_{n}$ and the total area covered by the rectangles is maximized \cite{ibmpuzzle}, \cite{Winkler2007}, \cite{Winkler2010a}, \cite{Winkler2010b}. The longstanding conjecture has been that at least half of $U$ can be covered when such rectangles are properly placed. In this paper, we give an existential proof of the conjecture.

Perfect packing of squares

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…

PackIt! 游戏化矩形装箱问题

PackIt! Gamified Rectangle Packing

本研究提出并分析了回合制矩形装箱游戏PackIt!，推导了完美装箱的存在条件，提出含新型SAT编码技术的自动推理方法可求解n≤50的完美对局，并证明了该问题的NP-hard性。

将倒数矩形的尾部装入等面积正方形

Packing Tails of Reciprocal Rectangles into Squares of Equal Area

该论文证明了Meir-Moser矩形打包问题的尾部版本：存在阈值m0，使得从m0开始的倒数矩形族可平移旋转装入边长为m^{-1/2}的正方形，面积恰好相等，并给出六步构造证明，经Lean 4验证。

通过底左算法实现 strip 包装的 13/6 近似算法

A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm

本文提出了一种新的矩形排序方式，利用底左算法在 strip 包装问题中实现了 13/6 的近似比。

分离相近元素的排列，以及环面上的矩形装填

Permutations that separate close elements, and rectangle packings in the torus

本文研究将圆上相近元素映射到相距足够远元素的 $(s,k)$-无冲突排列，并将其等价于环面矩形装填，最终确定所有参数下最大阈值 $σ(n,k)$。

Compactness Theorems for Geometric Packings

Moser asked whether the collection of rectangles of dimensions 1 x 1/2, 1/2 x 1/3, 1/3 x 1/4, ..., whose total area equals 1, can be packed into the unit square without overlap, and whether the collection of squares of side lengths 1/2, 1/3, 1/4, ... can be packed without overlap into a rectangle of area pi^2/6-1. Computational investigations have been made into packing these collections into squares of side length 1+epsilon and rectangles of area pi^2/6-1+epsilon, respectively, and one can consider the apparently weaker question whether such packings are possible for every positive number eps…

A computational algebraic geometry approach to classify partial Latin rectangles

This paper provides an in-depth analysis of how computational algebraic geometry can be used to deal with the problem of counting and classifying $r\times s$ partial Latin rectangles based on $n$ symbols of a given size, shape, type or structure. The computation of Hilbert functions and triangular systems of radical ideals enables us to solve this problem for all $r,s,n\leq 6$. As a by-product, explicit formulas are determined for the number of partial Latin rectangles of size up to six. We focus then on the study of non-compressible regular partial Latin squares and their equivalent incidence…

Exact Solution Techniques for Two-dimensional Cutting and Packing

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.

条形装箱问题的左下角算法

The Bottom-Left Algorithm for the Strip Packing Problem

本文研究条形装箱问题的左下角算法，将其近似比下界从5/4改进到4/3-ε，并证明该下界对正方形也成立，同时给出最坏顺序下的常数上界和10/3-ε下界，还分析了基于置换的局部搜索算法的近似性能。

用于 Meir-Moser 装填问题的 Slack-Pack 算法

Slack-Pack algorithm for Meir-Moser packing problem

针对 Meir-Moser 矩形铺砌问题，本文提出允许零件堆间留隙的 Slack-Pack 算法，以论证 Tao 关于大索引矩形可铺满等面积正方形的结果可推广至 t=1。

Enumerating partial Latin rectangles

This paper deals with distinct computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leq 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathr…

方形平铺算术的代数方法

An Algebraic Approach to the Arithmetic of Square Tilings

本文提出一种代数方法，通过块矩阵编码平铺几何，利用区间矩阵的全幺模性证明二维方形平铺分母最小公倍数以2^n为界，并推广到d维得到d^n界，同时简洁证明Kenyon定理。

正交复形阿波罗尼乌斯填充中的链环

Links in orthoplicial Apollonian packings

本文建立了阿波罗尼乌斯填充与纽结理论的联系，通过引入正则晶体球面填充切触图中链环的新表示，证明了任意代数链环可在正交复形阿波罗尼乌斯填充的立方截面中实现，并借此改进交错代数链环球数上界及揭示丢番图方程的几何本原解。

Exact Line Packings from Numerical Solutions

Recent progress in Zauner's conjecture has leveraged deep conjectures in algebraic number theory to promote numerical line packings to exact and verifiable solutions to the line packing problem. We introduce a numerical-to-exact technique in the real setting that does not require such conjectures. Our approach is completely reproducible, matching Sloane's database of putatively optimal numerical line packings with Mathematica's built-in implementation of cylindrical algebraic decomposition. As a proof of concept, we promote a putatively optimal numerical packing of eight points in the real pro…

Consequences of the packing problem

We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the validity of its consequences. Our work prompts additional questions and conjectures, which are presented together with their motivation.

重审Chazelle的Bottom-Left启发式实现：一个修正且严格的分析

Revisiting Chazelle's Implementation of the Bottom-Left Heuristic: A Corrected and Rigorous Analysis

本文修正并严格化Chazelle于1983年提出的Bottom-Left启发式实现，指出原运行时间分析在特定情况下会退化为Ω(n^3)，并通过补充形式化细节和修复缺陷，最终确认该实现具有O(n^2)时间复杂度和O(n)空间复杂度。

Packings in real projective spaces

This paper applies techniques from algebraic and differential geometry to determine how to best pack points in real projective spaces. We present a computer-assisted proof of the optimality of a particular 6-packing in $\mathbb{R}\mathbf{P}^3$, we introduce a linear-time constant-factor approximation algorithm for packing in the so-called Gerzon range, and we provide local optimality certificates for two infinite families of packings. Finally, we present perfected versions of various putatively optimal packings from Sloane's online database, along with a handful of infinite families they sugge…

Shortest (A+B)-path packing via hafnian

Björklund and Husfeldt developed a randomized polynomial time algorithm to solve the shortest two disjoint paths problem. Their algorithm is based on computation of permanents modulo 4 and the isolation lemma. In this paper, we consider the following generalization of the shortest two disjoint paths problem, and develop a similar algebraic algorithm. The shortest perfect $(A+B)$-path packing problem is: given an undirected graph $G$ and two disjoint node subsets $A,B$ with even cardinalities, find a shortest $|A|/2+|B|/2$ disjoint paths whose ends are both in $A$ or both in $B$. Besides its NP…

通过代数关联结构在有限群下进行集合打包

Packing sets under finite groups via algebraic incidence structures

本文研究了有限群作用下集合打包问题，通过代数关联结构方法，建立了轨道并集的下界，并在非集中条件下改进了结果。

Khovanov-Rozansky homology and Directed Cycles

We determine the cycle packing number of a directed graph using elementary projective algebraic geometry. Our idea is rooted in the Khovanov-Rozansky theory. In fact, using the Khovanov-Rozansky homology of a graph, we also obtain algebraic methods of detecting directed and undirected cycles containing a particular vertex or edge.

On local search and LP and SDP relaxations for k-Set Packing

Set packing is a fundamental problem that generalises some well-known combinatorial optimization problems and knows a lot of applications. It is equivalent to hypergraph matching and it is strongly related to the maximum independent set problem. In this thesis we study the k-set packing problem where given a universe U and a collection C of subsets over U, each of cardinality k, one needs to find the maximum collection of mutually disjoint subsets. Local search techniques have proved to be successful in the search for approximation algorithms, both for the unweighted and the weighted version o…