Computational Geometry-Theory and Applications最新文献

{"title":"Simple linear time algorithms for piercing pairwise intersecting disks","authors":"Ahmad Biniaz ,&nbsp;Prosenjit Bose ,&nbsp;Yunkai Wang","doi":"10.1016/j.comgeo.2023.102011","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102011","url":null,"abstract":"<div><p>A set <span><math><mi>D</mi></math></span> of disks in the plane is said to be pierced by a point set <em>P</em> if each disk in <span><math><mi>D</mi></math></span> contains a point of <em>P</em>. Any set of pairwise intersecting unit disks can be pierced by 3 points (Hadwiger and Debrunner (1955) <span>[7]</span>). Stachó and independently Danzer established that any set of pairwise intersecting arbitrary disks can be pierced by 4 points (Stachó (1981–1984) <span>[16]</span>. Danzer (1986) <span>[4]</span><span>). Existing linear-time algorithms for finding a set of 4 or 5 points that pierce pairwise intersecting disks of arbitrary radius use the LP-type problem as a subroutine. We present simple linear-time algorithms for finding 3 points for piercing pairwise intersecting unit disks, and 5 points for piercing pairwise intersecting disks of arbitrary radius. Our algorithms use simple geometric transformations and avoid heavy machinery. We also show that 3 points are sometimes necessary for piercing pairwise intersecting unit disks.</span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"114 ","pages":"Article 102011"},"PeriodicalIF":0.6,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multi-robot motion planning for unit discs with revolving areas","authors":"Pankaj K. Agarwal ,&nbsp;Tzvika Geft ,&nbsp;Dan Halperin ,&nbsp;Erin Taylor","doi":"10.1016/j.comgeo.2023.102019","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102019","url":null,"abstract":"<div><p>We study the problem of motion planning for a collection of <em>n</em> labeled unit disc robots in a polygonal environment. We assume that the robots have <em>revolving areas</em> around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a <em>weakly-monotone</em> motion plan, in which robots move according to an ordering as follows: during the turn of a robot <em>R</em> in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As <em>R</em> passes through a revolving area, a robot <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span>-approximation algorithm.</p><p><span>On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an </span><span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span><span> factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time </span><span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-approximation algorithm for this problem.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"114 ","pages":"Article 102019"},"PeriodicalIF":0.6,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49830655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cut locus realizations on convex polyhedra","authors":"Joseph O'Rourke ,&nbsp;Costin Vîlcu","doi":"10.1016/j.comgeo.2023.102010","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102010","url":null,"abstract":"<div><p>We prove that every positively weighted tree <em>T</em> can be realized as the cut locus <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> of a point <em>x</em><span> on a convex polyhedron </span><em>P</em>, with <em>T</em> edge weights matching <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> edge lengths. If <em>T</em> has <em>n</em> leaves, <em>P</em> has (in general) <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span><span> vertices. We show there is in fact a continuum of polyhedra </span><em>P</em> each realizing <em>T</em> for some <span><math><mi>x</mi><mo>∈</mo><mi>P</mi></math></span>. Three main tools in the proof are properties of the star unfolding of <em>P</em>, Alexandrov's gluing theorem, and a new cut-locus partition lemma. The construction of <em>P</em> from <em>T</em> is surprisingly simple.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"114 ","pages":"Article 102010"},"PeriodicalIF":0.6,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49790334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Augmenting graphs to minimize the radius","authors":"Joachim Gudmundsson ,&nbsp;Yuan Sha","doi":"10.1016/j.comgeo.2023.101996","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101996","url":null,"abstract":"<div><p>We study the problem of augmenting a metric graph by adding <em>k</em> edges while minimizing the radius of the augmented graph. We give a simple 3-approximation algorithm and show that there is no polynomial-time <span><math><mo>(</mo><mn>5</mn><mo>/</mo><mn>3</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo></math></span>-approximation algorithm, for any <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span>, unless <span><math><mi>P</mi><mo>=</mo><mi>N</mi><mi>P</mi></math></span>.</p><p>We also give two exact algorithms for the special case when the input graph is a tree, one of which is generalized to handle metric graphs with bounded treewidth.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"113 ","pages":"Article 101996"},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49845789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Keep your distance: Land division with separation","authors":"Edith Elkind ,&nbsp;Erel Segal-Halevi ,&nbsp;Warut Suksompong","doi":"10.1016/j.comgeo.2023.102006","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102006","url":null,"abstract":"<div><p>This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of <em>proportionality</em> is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the <em>ordinal maximin share approximation</em>, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"113 ","pages":"Article 102006"},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49845790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Any platonic solid can transform to another by O(1) refoldings","authors":"Erik D. Demaine,&nbsp;Martin L. Demaine,&nbsp;Yevhenii Diomidov,&nbsp;Tonan Kamata,&nbsp;Ryuhei Uehara,&nbsp;Hanyu Alice Zhang","doi":"10.1016/j.comgeo.2023.101995","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101995","url":null,"abstract":"<div><p><span>We show that several classes of polyhedra are joined by a sequence of </span><span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span><span> refolding steps, where each refolding step unfolds the current polyhedron (allowing cuts anywhere on the surface and allowing overlap) and folds that unfolding into exactly the next polyhedron; in other words, a polyhedron is refoldable into another polyhedron if they share a common unfolding. Specifically, assuming equal surface area, we prove that (1) any two tetramonohedra are refoldable to each other, (2) any doubly covered triangle is refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and doubly covered regular polygon<span> is refoldable to a tetramonohedron, (4) any tetrahedron<span> has a 3-step refolding sequence to a tetramonohedron, and (5) the regular dodecahedron<span> has a 4-step refolding sequence to a tetramonohedron. In particular, we obtain a ≤6-step refolding sequence between any pair of Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all other Platonic solids. As far as the authors know, this is the first result about common unfolding involving the regular dodecahedron.</span></span></span></span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"113 ","pages":"Article 101995"},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49845788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometric dominating-set and set-cover via local-search","authors":"Minati De ,&nbsp;Abhiruk Lahiri","doi":"10.1016/j.comgeo.2023.102007","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.102007","url":null,"abstract":"<div><p>In this paper, we study two classic optimization problems<span>: minimum geometric dominating set and set cover. In the dominating-set problem, for a given set of objects in the plane as input, the objective is to choose a minimum number of input objects such that every input object is dominated by the chosen set of objects. Here, we say that one object is dominated by another if their intersection is nonempty. For the second problem, for a given set of points and objects in the plane, the objective is to choose a minimum number of objects to cover all the points. This is a particular version of the set-cover problem.</span></p><p>Both problems have been well-studied, subject to various restrictions on the input objects. These problems are <span><math><mi>APX</mi></math></span><span>-hard for object sets consisting of axis-parallel rectangles, ellipses, </span><em>α</em><span>-fat objects of constant description complexity, and convex polygons. On the other hand, </span><span><math><mi>PTAS</mi></math></span><span>s (polynomial time approximation schemes) are known for object sets consisting of disks or unit squares. Surprisingly, a </span><span><math><mi>PTAS</mi></math></span> was unknown even for arbitrary squares. For both problems obtaining a <span><math><mi>PTAS</mi></math></span> remains open for a large class of objects.</p><p>For the dominating-set problem, we prove that a popular local-search algorithm leads to a <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></math></span> approximation for a family of homothets of a convex object (which includes arbitrary squares, <em>k</em><span>-regular polygons, translated and scaled copies of a convex set, etc.) in </span><span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></msup></math></span> time. On the other hand, the same approach leads to a <span><math><mi>PTAS</mi></math></span><span> for the geometric covering problem<span> when the objects are convex pseudodisks (which include disks, unit height rectangles, homothetic convex objects, etc.). Consequently, we obtain an easy-to-implement approximation algorithm for both problems for a large class of objects, significantly improving the best-known approximation guarantees.</span></span></p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"113 ","pages":"Article 102007"},"PeriodicalIF":0.6,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Angles of arc-polygons and Lombardi drawings of cacti","authors":"David Eppstein,&nbsp;Daniel Frishberg,&nbsp;Martha C. Osegueda","doi":"10.1016/j.comgeo.2023.101982","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101982","url":null,"abstract":"<div><p>We characterize the triples of interior angles that are possible in non-self-crossing triangles with circular-arc sides, and we prove that a given cyclic sequence of angles can be realized by a non-self-crossing polygon with circular-arc sides whenever all angles are ≤<em>π</em>. As a consequence of these results, we prove that every cactus has a planar Lombardi drawing (a drawing with edges depicted as circular arcs, meeting at equal angles at each vertex) for its natural embedding in which every cycle of the cactus is a face of the drawing. However, there exist planar embeddings of cacti that do not have planar Lombardi drawings.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101982"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bottleneck matching in the plane","authors":"Matthew J. Katz ,&nbsp;Micha Sharir","doi":"10.1016/j.comgeo.2023.101986","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101986","url":null,"abstract":"<div><p><span>We present a randomized algorithm that with high probability finds a bottleneck matching in a set of </span><span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>ℓ</mi></math></span> points in the plane. The algorithm's running time is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ω</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><mi>ω</mi><mo>&gt;</mo><mn>2</mn></math></span> is a constant such that any two <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices can be multiplied in time <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span>. The state of the art in fast matrix multiplication allows us to set <span><math><mi>ω</mi><mo>=</mo><mn>2.3728596</mn></math></span>.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101986"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the geometric priority set cover problem","authors":"Aritra Banik ,&nbsp;Rajiv Raman ,&nbsp;Saurabh Ray","doi":"10.1016/j.comgeo.2023.101984","DOIUrl":"https://doi.org/10.1016/j.comgeo.2023.101984","url":null,"abstract":"<div><p><span>We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor </span>approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.</p></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"112 ","pages":"Article 101984"},"PeriodicalIF":0.6,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49795351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}