Discrete & Computational Geometry最新文献_第9页

Geometric Stabbing via Threshold Rounding and Factor Revealing LPs 通过阈值舍入和因子揭示lp的几何刺伤

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-11-27 DOI: 10.1007/s00454-023-00608-8

Khaled Elbassioni, Saurabh Ray

{"title":"Geometric Stabbing via Threshold Rounding and Factor Revealing LPs","authors":"Khaled Elbassioni, Saurabh Ray","doi":"10.1007/s00454-023-00608-8","DOIUrl":"https://doi.org/10.1007/s00454-023-00608-8","url":null,"abstract":"Kovaleva and Spieksma (SIAM J Discrete Math 20(3):48–768, 2006) considered the problem of stabbing a given set of horizontal line segments with the smallest number of horizontal and vertical lines. The standard LP relaxation for this problem is easily shown to have an integrality gap of at most 2 by treating the horizontal and vertical lines separately. However, Kovaleva and Spieksma observed that threshold rounding can be used to obtain an integrality gap of (e/(e-1) approx 1.58) which is also shown to be tight. This is one of the rare known examples where the obvious upper bound of 2 on the integrality gap of the standard LP relaxation can be improved. Our goal in this paper is to extend their proof to two other problems where the goal is to stab a set (mathcal {R}) of objects with horizontal and vertical lines: in the first problem (mathcal {R}) is a set of horizontal and vertical line segments, and in the second problem (mathcal {R}) is a set of unit sized squares. The proof of Kovaleva and Spieksma essentially shows the existence of an appropriate threshold which yields the improved approximation factor. We begin by showing that a random threshold picked from an appropriate distribution works. This reduces the problem to finding an appropriate distribution for a desired approximation ratio. In the first problem, we show that the required distribution can be found by solving a linear program. In the second problem, while it seems harder to find the optimal distribution, we show that using the uniform distribution an improved approximation factor can still be obtained by solving a number of linear programs.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"126 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507769","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Trilateration Using Unlabeled Path or Loop Lengths 使用未标记路径或循环长度的三边测量

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-11-25 DOI: 10.1007/s00454-023-00605-x

Ioannis Gkioulekas, Steven J. Gortler, Louis Theran, Todd Zickler

{"title":"Trilateration Using Unlabeled Path or Loop Lengths","authors":"Ioannis Gkioulekas, Steven J. Gortler, Louis Theran, Todd Zickler","doi":"10.1007/s00454-023-00605-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00605-x","url":null,"abstract":"Let (textbf{p}) be a configuration of n points in (mathbb R^d) for some n and some (d ge 2). Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing (textbf{p}) given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when (textbf{p}) will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that (textbf{p}) is generic.","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"72 9-10","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138507775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Ehrhart Quasi-Polynomials of Almost Integral Polytopes 几乎整多边形的Ehrhart拟多项式

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-11-24 DOI: 10.1007/s00454-023-00604-y

Christopher de Vries, Masahiko Yoshinaga

引用次数: 3

Volumes of Subset Minkowski Sums and the Lyusternik Region 子集Minkowski和的体积与Lyusternik域

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-11-21 DOI: 10.1007/s00454-023-00606-w

Franck Barthe, Mokshay Madiman

引用次数: 5

Grounded L-Graphs Are Polynomially $$chi $$ -Bounded 接地l -图是多项式$$chi $$有界的

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-11-16 DOI: 10.1007/s00454-023-00592-z

James Davies, Tomasz Krawczyk, Rose McCarty, Bartosz Walczak

引用次数: 3

Some New Results on Geometric Transversals 关于几何截线的一些新结果

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-11-16 DOI: 10.1007/s00454-023-00573-2

Otfried Cheong, Xavier Goaoc, Andreas F. Holmsen

引用次数: 1

The Visible-Volume Function of a Set of Cameras is Continuous, Piecewise Rational, Locally Lipschitz, and Semi-Algebraic in All Dimensions 一组相机的可见体积函数是连续的、分段有理的、局部Lipschitz的和全维半代数的

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-08-12 DOI: 10.1007/s00454-023-00541-w

Jörg Rambau

引用次数: 0

Self-Affine Tiles Generated by a Finite Number of Matrices 有限个矩阵生成的自仿射块

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-07-23 DOI: 10.1007/s00454-023-00529-6

Guotai Deng, Chuntai Liu, Sze-Man Ngai

引用次数: 0

Complete Characterization of Polyhedral Self-Affine Tiles 多面体自仿射瓷砖的完全表征

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-07-19 DOI: 10.1007/s00454-023-00527-8

V. Protasov, T. Zaitseva

引用次数: 0

Uniformly Acute Triangulations of PSLGs PSLG的一致锐角三角剖分

IF 0.8 3区数学

Discrete & Computational Geometry Pub Date : 2023-07-08 DOI: 10.1007/s00454-023-00524-x

C. Bishop

引用次数: 2