Jongmin Choi, Dahye Jeong, Hee-Kap Ahn
We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an $O(n\log n)$-time algorithm for the two-center problem for a convex polygon, where $n$ is the number of vertices of the polygon. This improves upon the previous best algorithm for the problem.
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In this note we prove that, if $S_n$ is the greatest area of a rectangle which can be covered with $n$ unit disks, then $2\leq S_n/n<3 \sqrt{3}/2$, and these are the best constants; moreover, for $Δ(n):=(3\sqrt{3}/2)n-S_n$, we have $0.727384<\liminfΔ(n)/\sqrt{n}<2.121321$ and $0.727384<\limsupΔ(n)/\sqrt{n}<4.165064$.
In this paper, we consider the problem of covering a plane region with unit discs. We present an improved upper bound and the first nontrivial lower bound on the number of discs needed for such a covering, depending on the area and perimeter of the region. We provide algorithms for efficient covering of convex polygonal regions using unit discs. We show that the computational complexity of the algorithms is pseudo-polynomial in the size of the input and the output. We also show that these algorithms provide a constant factor approximation of the optimal covering of the region.
Given a set of disjoint simple polygons $σ_1, \ldots, σ_n$, of total complexity $N$, consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by their common convex hull. This process continues until no pair of polygons intersect. We show that this process has a unique output, which is a cover of the input polygons by a set of disjoint convex polygons, of total minimum area. Furthermore, we present a near linear time algorithm for computing this partition. The more general problem of covering a set of $N$ segm…
We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k denoting the number centers found by the algorithm.
We give exact and approximation algorithms for two-center problems when the input is a set $\mathcal{D}$ of disks in the plane. We first study the problem of finding two smallest congruent disks such that each disk in $\mathcal{D}$ intersects one of these two disks. Then we study the problem of covering the set $\mathcal{D}$ by two smallest congruent disks.
Given a polygon $P$, for two points $s$ and $t$ contained in the polygon, their \emph{geodesic distance} is the length of the shortest $st$-path within $P$. A \emph{geodesic disk} of radius $r$ centered at a point $v \in P$ is the set of points in $P$ whose geodesic distance to $v$ is at most $r$. We present a polynomial time $2$-approximation algorithm for finding a densest geodesic unit disk packing in $P$. Allowing arbitrary radii but constraining the number of disks to be $k$, we present a $4$-approximation algorithm for finding a packing in $P$ with $k$ geodesic disks whose minimum radius…
Convex Covering Using Collections of Convex Polygons and Set Cover
针对带孔凸多边形的凸覆盖问题，本文提出先收集大凸多边形再求解多个集合覆盖问题的启发式算法，并在CG:SHOP 2023挑战赛中获得第二名。
Covering Simple Orthogonal Polygons with Rectangles
针对简单正交多边形的矩形覆盖问题，研究证明简单局部搜索算法可得到边界覆盖问题的PTAS，但该技术无法用于内部覆盖问题，同时其对偶最大反矩形问题存在较大局部性间隙。
Given a convex polygon $P$ with $n$ vertices, the two-center problem is to find two congruent closed disks of minimum radius such that they completely cover $P$. We propose an algorithm for this problem in the streaming setup, where the input stream is the vertices of the polygon in clockwise order. It produces a radius $r$ satisfying $r\leq2r_{opt}$ using $O(1)$ space, where $r_{opt}$ is the optimum solution. Next, we show that in non-streaming setup, we can improve the approximation factor by $r\leq 1.84 r_{opt}$, maintaining the time complexity of the algorithm to $O(n)$, and using $O(1)$ e…
Covering simple orthogonal polygons with $r$-stars
本研究解决简单正交多边形中的$r$-星覆盖问题（即带矩形视野的简单正交多边形点警卫问题），提出可在二次时间内完成求解的方法。
We study the problem of decomposing (i.e. partitioning and covering) polygons into components that are $α$-fat, which means that the aspect ratio of each subpolygon is at most $α$. We consider decompositions without Steiner points. We present a polynomial-time algorithm for simple polygons that finds the minimum $α$ such that an $α$-fat partition exists. Furthermore, we show that finding an $α$-fat partition or covering with minimum cardinality is NP-hard for polygons with holes.
Hardness of Packing, Covering and Partitioning Simple Polygons with Unit Squares
本文证明将单位正方形填充、覆盖和划分简单多边形（即使无孔且坐标半整数）均为NP困难，通过非平面几何实现Planar-3SAT归约，首次证明无孔多边形划分问题的NP困难性。
We prove that for any convex polygon $S$ with at least four sides, or a concave one with no parallel sides, and any $m>0$, there is an $m$-fold covering of the plane with homothetic copies of $S$ that cannot be decomposed into two coverings.
We present approximation algorithms with O(n^3) processing time for the minimum vertex and edge guard problems in simple polygons. It is improved from previous O(n^4) time algorithms of Ghosh. For simple polygon, there are O(n^3) visibility regions, thus any approximation algorithm for the set covering problem with approximation ratio of log(n) can be used for the approximation of n vertex and edge guard problems with O(n^3) visibility sequence. We prove that the visibility of all points in simple polygons is guaranteed by covering O(n^2) sinks from vertices and edges : It comes to O(n^3) time…
A planar set $P$ is said to be cover-decomposable if there is a constant $k=k(P)$ such that every $k$-fold covering of the plane with translates of $P$ can be decomposed into two coverings. It is known that open convex polygons are cover-decomposable. Here we show that closed, centrally symmetric convex polygons are also cover-decomposable. We also show that an infinite-fold covering of the plane with translates of $P$ can be decomposed into two infinite-fold coverings. Both results hold for coverings of any subset of the plane.
Efficient Exact Algorithms for Minimum Covering of Orthogonal Polygons with Squares
本文针对无孔正交多边形正方形覆盖(OPCS)问题，提出了运行时间分别为 $\mathcal O(n^{10})$ 和 $\mathcal O(n^2 + k^{10} \cdot n)$ 的多项式时间精确算法，并修正了带孔情况(OPCSH)的NP完全性证明。
We show that a $k$-fold covering using translates of an arbitrary convex polygon can be decomposed into $Ω(k)$ covers (using an efficient algorithm). We generalize this result to obtain a constant factor approximation to the sensor cover problem where the ranges of the sensors are translates of a given convex polygon. The crucial ingredient in this generalization is a constant factor approximation algorithm for a one-dimensional version of the sensor cover problem, called the Restricted Strip Cover (RSC) problem, where sensors are intervals of possibly different lengths. Our algorithm for RSC …
Covering Rectilinear Polygons with Area-Weighted Rectangles
针对带孔直角多边形的面积加权矩形最小总权重覆盖NP难问题，发现最优解新性质，改进相关算法并提出新近似与启发式方法，大规模实验验证其速度快且解接近最优。
Decomposing a Simple Polygon with Geodesic Unit-Balls
研究如何用\(\ell_2\)度量将简单多边形覆盖和分割为具有单位测地半径或直径的部分，针对半径问题给出覆盖和分割的9近似算法，针对直径问题的分割版本给出15近似算法，改进了前人复杂算法。
Covering and Partitioning Complex Objects with Small Pieces
研究如何用最少的小块覆盖或划分多边形，证明覆盖和划分问题等价，提出一种局部搜索算法获得PTAS，同时指出三维情况下近似最优解为NP难问题。
Motivated by a new way of visualizing hypergraphs, we study the following problem. Consider a rectangular grid and a set of colors $χ$. Each cell $s$ in the grid is assigned a subset of colors $χ_s \subseteq χ$ and should be partitioned such that for each color $c\in χ_s$ at least one piece in the cell is identified with $c$. Cells assigned the empty color set remain white. We focus on the case where $χ= \{\text{red},\text{blue}\}$. Is it possible to partition each cell in the grid such that the unions of the resulting red and blue pieces form two connected polygons? We analyze the combinatori…
In the MINIMUM CONVEX COVER (MCC) problem, we are given a simple polygon $\mathcal P$ and an integer $k$, and the question is if there exist $k$ convex polygons whose union is $\mathcal P$. It is known that MCC is $\mathsf{NP}$-hard [Culberson & Reckhow: Covering polygons is hard, FOCS 1988/Journal of Algorithms 1994] and in $\exists\mathbb{R}$ [O'Rourke: The complexity of computing minimum convex covers for polygons, Allerton 1982]. We prove that MCC is $\exists\mathbb{R}$-hard, and the problem is thus $\exists\mathbb{R}$-complete. In other words, the problem is equivalent to deciding whether…
We prove that for every centrally symmetric convex polygon Q, there exists a constant alpha such that any alpha*k-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Toth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery lifetime.
A covering path for a planar point set is a path drawn in the plane with straight-line edges such that every point lies at a vertex or on an edge of the path. A covering tree is defined analogously. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane can be covered by a noncrossing path with at most $π(n)$ edges. Let $τ(n)$ be the analogous number for noncrossing covering trees. Dumitrescu, Gerbner, Keszegh, and Tóth (Discrete & Computational Geometry, 2014) established the following inequalities: \[\frac{5n}{9} - O(1) < π(n) < \left(1-\frac{1}{601080391}\right)n, \…
We give an overview of the 2023 Computational Geometry Challenge targeting the problem Minimum Coverage by Convex Polygons, which consists of covering a given polygonal region (possibly with holes) by a minimum number of convex subsets, a problem with a long-standing tradition in Computational Geometry.
Given a polygon and a visibility range, the Myopic Watchman Problem with Discrete Vision (MWPDV) asks for a closed path P and a set of scan points S, such that (i) every point of the polygon is within visibility range of a scan point; and (ii) path length plus weighted sum of scan number along the tour is minimized. Alternatively, the bicriteria problem (ii') aims at minimizing both scan number and tour length. We consider both lawn mowing (in which tour and scan points may leave P) and milling (in which tour, scan points and visibility must stay within P) variants for the MWPDV; even for simp…
Super Guarding and Dark Rays in Art Galleries
研究艺术画廊变体中守卫不能看透彼此且需覆盖k次的问题，给出凸多边形k-cover的紧致界并证明其非平凡性，同时为简单多边形提供界。
This paper discusses the problem of covering and hitting a set of line segments $\cal L$ in ${\mathbb R}^2$ by a pair of axis-parallel squares such that the side length of the larger of the two squares is minimized. We also discuss the restricted version of covering, where each line segment in $\cal L$ is to be covered completely by at least one square. The proposed algorithm for the covering problem reports the optimum result by executing only two passes of reading the input data sequentially. The algorithm proposed for the hitting and restricted covering problems produces optimum result in $…
Can folding a piece of paper flat make it larger? We explore whether a shape $S$ must be scaled to cover a flat-folded copy of itself. We consider both single folds and arbitrary folds (continuous piecewise isometries $S\rightarrow R^2$). The underlying problem is motivated by computational origami, and is related to other covering and fixturing problems, such as Lebesgue's universal cover problem and force closure grasps. In addition to considering special shapes (squares, equilateral triangles, polygons and disks), we give upper and lower bounds on scale factors for single folds of convex ob…
We present filling as a new type of spatial subdivision problem that is related to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In n-dimensional space, if the objects are polydisperse n-balls, we show that solutions correspond to sets of maximal n-balls and the solution space can reduced to the medial axis of a shape. We examine the structure of the solution space in two dimensions. For the filling of polygons, we provide detailed descriptions of a heuristic and a genetic a…
A fast improved quasi-physical dynamic algorithm for efficient wireless coverage in convex polygonal regions
针对凸多边形无线节点部署问题，提出改进类物理动态算法IQPD，通过保结构初始化、精细化虚拟力模型与边界环绕策略，实现更高的覆盖率与节点利用率，性能优于多种元启发式算法。
Minimum Cost Polygon Overlay (MCPO) is a unique two-dimensional optimization problem that involves the task of covering a polygon shaped area with a series of rectangular shaped panels. This has a number of applications in the construction industry. This work examines the MCPO problem in order to construct a model that captures essential parameters of the problem to be solved automatically using numerical optimization algorithms. Three algorithms have been implemented of the actual optimization task: the greedy search, the Monte Carlo (MC) method, and the Genetic Algorithm (GA). Results are pr…
We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opp…
Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number $k$ of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given $Δ$ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classica…
Minimum Star Partitions of Simple Polygons in Polynomial Time
本文提出了一种多项式时间算法，用于将简单多边形划分为最少数量的星形多边形，解决了长期存在的开放问题。
In this paper, we consider the problem of partitioning a polygon into a set of connected disjoint sub-polygons, each of which covers an area of a specific size. The work is motivated by terrain covering applications in robotics, where the goal is to find a set of efficient plans for a team of heterogeneous robots to cover a given area. Within this application, solving a polygon partitioning problem is an essential stepping stone. Unlike previous work, the problem formulation proposed in this paper also considers a compactness metric of the generated sub-polygons, in addition to the area size c…
We consider the planar Euclidean two-center problem in which given $n$ points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic $O(n \log n)$-time algorithm for the case that the centers of the two optimal disks are close to each other, that is, the overlap of the two optimal disks is a constant fraction of the disk area. We also present a deterministic $O(n\log n)$-time algorithm for the case that the input points are in convex position. Both results improve the previous best $O(n\log n\log\log n)$ bound on the problems.
The local search framework for obtaining PTASs for NP-hard geometric optimization problems was introduced, independently, by Chan and Har-Peled (2009) and Mustafa and Ray (2010). In this paper, we generalize the framework by extending its analysis to additional families of graphs, beyond the family of planar graphs. We then present several applications of the generalized framework, some of which are very different from those presented to date (using the original framework). These applications include PTASs for finding a maximum l-shallow set of a set of fat objects, for finding a maximum trian…
Peeling Rotten Potatoes for a Faster Approximation of Convex Cover
本文提出一种新方法，通过离散化问题并将其转化为集合覆盖问题，以在降低运行时间的同时保持O(log n)的近似保证，解决了凸覆盖问题。
Providing robots with large-scale robot skin is a challenging goal, especially when considering surfaces characterized by different shapes and curvatures. The problem originates from technological advances in tactile sensing, and in particular from two requirements: (i) covering the largest possible area of a robot's surface with tactile sensors, and (ii) doing it using cheap, replicable hardware modules. Given modules of a specific shape, the problem of optimally placing them requires to maximize the number of modules that can be fixed on the selected robot body part. Differently from previou…
Bicriteria Polygon Aggregation with Arbitrary Shapes
研究用不相交的代表区域覆盖多边形集合以进行聚类的问题，目标是最小化区域总面积与总周长的加权和，提出一种时间复杂度为Õ(n^4)的算法，并刻画最优解边界由输入边和等半径圆弧组成。
The Contiguous Art Gallery Problem is Solvable in Polynomial Time
本文研究连续艺术馆问题，提出一种时间复杂度为 $\mathcal{O}(n^6 \log n)$ 的贪心多项式时间 RAM 算法，证明该问题属于 P，并为受限环境提供额外算法。
We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R^3 can be colored with k colors so that every translate of the negative octant containing at least k^6 points contains at least one of each color. The best previously known bound was doubly exponential in k. This yields, among other corollaries, the first polynomial bound for the decomposability of multiple coverings by homothetic triangles. We also investigate related decomposition pr…
A Certified Lower Bound for Lebesgue's Universal Cover Problem
通过结合几何估计与计算机搜索，Brass和Sharifi证明了Lebesgue通用覆盖问题凸形式的下界0.832；本文给出了该计算基于证书的复现，包含有限自适应账本、终端路径重放、三个局部下界证书族、大型表格的紧凑完整性审计以及将重放数据与下界陈述连接的有限证明义务层。
Let P be a set of n points in R^3. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n^3 log^5 n) expected time, and the second algorithm runs in O((n^2 log^5 n) /(1-r*/r_0)^3) expected time, where r* is the radius of the 2-center balls of P and r_0 is the radius of the smallest enclosing ball of P. The second algorithm is faster than the first one as long as r* is not too close to r_0, which is equivalent to the condition that the centers of the two…
We give the first algorithmic study of a class of ``covering tour'' problems related to the geometric Traveling Salesman Problem: Find a polygonal tour for a cutter so that it sweeps out a specified region (``pocket''), in order to minimize a cost that depends mainly on the number of em turns. These problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems, as well as arc routing (``postman'') problems with turn penalties. We prove the NP-completeness of minimum-turn milling and give efficient approximati…
Curves of constant width and Lebesgue's covering problem
本文针对勒贝格覆盖问题，以等宽曲线为测试集，证明凸通用覆盖集面积下界为0.8344，改进了此前结果，相关证明含解析归约与有限计算。
Perfectly Guarding Straits: Exact Algorithms for Weak Visibility Polygons
针对弱可见多边形，提出精确算法求解海峡守卫问题，将二维覆盖化为一维，时间复杂度为输出敏感，并改进海拔地形守卫的已知界。
On Minimum Aerial Photographs for Planar Region Coverage: Hardness and Approximation
研究用正方形和圆形覆盖简单平面多边形的计算复杂性，证明了不可近似性间隙并开发了2.828-最优近似算法。
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