Pub Date : 2025-11-19DOI: 10.1016/j.comgeo.2025.102232
Jaehoon Chung , Sang Won Bae , Chan-Su Shin , Sang Duk Yoon , Hee-Kap Ahn
We consider two optimization problems of approximating a convex polygon in the plane, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation θ is a copy of an axis-aligned histogon rotated by θ in a counterclockwise direction. Our goal is to compute a largest inscribed histogon and a smallest circumscribed histogon over all orientations in . Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms for these versions of the inscribed histogon problem. For the circumscribed histogon problem, we present an efficient algorithm whose running time depends on the diameter and the number of vertices of the input polygon. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.
{"title":"Inscribed and circumscribed histogons of a convex polygon","authors":"Jaehoon Chung , Sang Won Bae , Chan-Su Shin , Sang Duk Yoon , Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102232","DOIUrl":"10.1016/j.comgeo.2025.102232","url":null,"abstract":"<div><div>We consider two optimization problems of approximating a convex polygon in the plane, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation <em>θ</em> is a copy of an axis-aligned histogon rotated by <em>θ</em> in a counterclockwise direction. Our goal is to compute a largest inscribed histogon and a smallest circumscribed histogon over all orientations in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>)</mo></math></span>. Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms for these versions of the inscribed histogon problem. For the circumscribed histogon problem, we present an efficient algorithm whose running time depends on the diameter and the number of vertices of the input polygon. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102232"},"PeriodicalIF":0.7,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571941","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}
Pub Date : 2025-11-03DOI: 10.1016/j.comgeo.2025.102231
Sajjad Hashemian , Mohammad Saeed Arvenaghi , Ebrahim Ardeshir-Larijani
In this paper, we introduce new algorithms for Principal Component Analysis (PCA) with outliers. Utilizing techniques from computational geometry, specifically higher-degree Voronoi diagrams, we navigate to the optimal subspace for PCA even in the presence of outliers. This approach achieves an optimal solution with time complexity of . Additionally, we present a randomized algorithm with complexity . Our approach leverages properties of high-dimensional spaces and the separation condition of outliers to efficiently recover the optimal subspace. Our results demonstrate that higher-degree Voronoi diagrams, combined with probabilistic subspace selection techniques, provide an effective and scalable solution for PCA with outliers.
{"title":"Optimal bound for PCA with outliers using higher-degree Voronoi diagrams","authors":"Sajjad Hashemian , Mohammad Saeed Arvenaghi , Ebrahim Ardeshir-Larijani","doi":"10.1016/j.comgeo.2025.102231","DOIUrl":"10.1016/j.comgeo.2025.102231","url":null,"abstract":"<div><div>In this paper, we introduce new algorithms for Principal Component Analysis (PCA) with outliers. Utilizing techniques from computational geometry, specifically higher-degree Voronoi diagrams, we navigate to the optimal subspace for PCA even in the presence of outliers. This approach achieves an optimal solution with time complexity of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mtext>poly</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>. Additionally, we present a randomized algorithm with complexity <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>log</mi><mo></mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>δ</mi><mo>)</mo><mo>/</mo><mi>C</mi><mo>(</mo><mi>d</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>α</mi><mo>)</mo><mo>)</mo><mtext>poly</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>d</mi><mo>)</mo></math></span>. Our approach leverages properties of high-dimensional spaces and the separation condition of outliers to efficiently recover the optimal subspace. Our results demonstrate that higher-degree Voronoi diagrams, combined with probabilistic subspace selection techniques, provide an effective and scalable solution for PCA with outliers.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102231"},"PeriodicalIF":0.7,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467266","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}
Pub Date : 2025-09-25DOI: 10.1016/j.comgeo.2025.102230
Ahmad Biniaz, Mohammad Hashemi
We study two problems related to the city guarding and the art gallery problems.
1.
Given a city with k rectangular buildings, we prove that cameras of field of view are always sufficient to guard the free space (the ground, walls, roofs, and the sky). This answers a conjecture of Daescu and Malik (2020) [7].
2.
Given k orthogonally convex polygons of total m vertices in the plane, we prove that cameras of field of view are always sufficient to guard the free space (avoiding all the polygons). This answers another conjecture of Daescu and Malik (2021) [8].
Both upper bounds are tight in the sense that there are input instances that require these many cameras. Our proofs are constructive and suggest simple polynomial-time algorithms for placing these many cameras.
We then generalize the above bounds to arbitrary convex-shape buildings. We can guard the free space of k buildings of total size m by cameras. For k simple polygons with c convex vertices in the plane we can guard the free space by cameras. Again, both these bounds are tight.
{"title":"City guarding with cameras of bounded field of view","authors":"Ahmad Biniaz, Mohammad Hashemi","doi":"10.1016/j.comgeo.2025.102230","DOIUrl":"10.1016/j.comgeo.2025.102230","url":null,"abstract":"<div><div>We study two problems related to the city guarding and the art gallery problems.<ul><li><span>1.</span><span><div>Given a city with <em>k</em> rectangular buildings, we prove that <span><math><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras of <span><math><msup><mrow><mn>180</mn></mrow><mrow><mo>∘</mo></mrow></msup></math></span> field of view are always sufficient to guard the free space (the ground, walls, roofs, and the sky). This answers a conjecture of Daescu and Malik (2020) <span><span>[7]</span></span>.</div></span></li><li><span>2.</span><span><div>Given <em>k</em> orthogonally convex polygons of total <em>m</em> vertices in the plane, we prove that <span><math><mfrac><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras of <span><math><msup><mrow><mn>180</mn></mrow><mrow><mo>∘</mo></mrow></msup></math></span> field of view are always sufficient to guard the free space (avoiding all the polygons). This answers another conjecture of Daescu and Malik (2021) <span><span>[8]</span></span>.</div></span></li></ul> Both upper bounds are tight in the sense that there are input instances that require these many cameras. Our proofs are constructive and suggest simple polynomial-time algorithms for placing these many cameras.</div><div>We then generalize the above bounds to arbitrary convex-shape buildings. We can guard the free space of <em>k</em> buildings of total size <em>m</em> by <span><math><mi>m</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras. For <em>k</em> simple polygons with <em>c</em> convex vertices in the plane we can guard the free space by <span><math><mi>c</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span> cameras. Again, both these bounds are tight.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102230"},"PeriodicalIF":0.7,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222355","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}
Pub Date : 2025-09-18DOI: 10.1016/j.comgeo.2025.102229
Péter Ágoston , Adrian Dumitrescu , Arsenii Sagdeev , Karamjeet Singh , Ji Zeng
For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of n points in , there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least . Apart from the factor, this bound is the best possible. As for the abstract setting, we show that for every n-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree .
{"title":"Maximizing the maximum degree in ordered nearest neighbor graphs","authors":"Péter Ágoston , Adrian Dumitrescu , Arsenii Sagdeev , Karamjeet Singh , Ji Zeng","doi":"10.1016/j.comgeo.2025.102229","DOIUrl":"10.1016/j.comgeo.2025.102229","url":null,"abstract":"<div><div>For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the <em>ordered Nearest Neighbor Graph</em> is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of <em>n</em> points in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least <span><math><mi>log</mi><mo></mo><mi>n</mi><mo>/</mo><mo>(</mo><mn>4</mn><mi>d</mi><mo>)</mo></math></span>. Apart from the <span><math><mn>1</mn><mo>/</mo><mo>(</mo><mn>4</mn><mi>d</mi><mo>)</mo></math></span> factor, this bound is the best possible. As for the abstract setting, we show that for every <em>n</em>-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree <span><math><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo></mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi></mrow></msqrt><mo>)</mo></math></span>.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102229"},"PeriodicalIF":0.7,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145120134","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}
Pub Date : 2025-09-12DOI: 10.1016/j.comgeo.2025.102228
C.R. Subramanian
<div><div>We present a generic formulation of an algorithmic paradigm for approximating maximum weighted independent sets (MWIS) in arbitrary vertex weighted graphs. A special case of this paradigm has been proposed earlier for geometric intersection graphs. Here, we propose and analyse a much more general formulation. As part of this formulation, we introduce a new graph parameter which plays a role in bounding the approximation factor of the algorithms. By applying this paradigm to intersection graph classes of specific types of geometric objects, we obtain efficient algorithms which approximate a MWIS within <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> multiplicative factors. It is also shown that the same approach can be generalised to obtain efficient approximation algorithms for computing an optimal weight <span><math><mi>P</mi></math></span>-subgraphs where <span><math><mi>P</mi></math></span> is a suitable hereditary property.</div><div>Applying our paradigm, we establish, for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>]</mo></math></span>, that MWIS of the intersection graph of a given collection of weighted <em>k</em>-dimensional <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> spheres (having a common radius) can be efficiently approximated within a multiplicative factor of <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. The running time can be brought down to <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span> at the cost of increasing the approximation guarantee to <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, for some constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> depending only on <em>p</em> and <em>k</em>. It is also shown that the above MWIS-approximation results can be extended to MWIS-approximation over the more general intersection graphs of finite collections of connected, full-dimensional and centrally-symmetric bodies in <em>k</em>-dimensional, <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-spaces, for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>]</mo></math></span>.</div><div>In a related development, we also establish the following graph theoretic result which will be of independent interest: For every <span><math><mi>p</mi><mo>∈</
{"title":"Approximation of MWIS on geometric intersection graphs","authors":"C.R. Subramanian","doi":"10.1016/j.comgeo.2025.102228","DOIUrl":"10.1016/j.comgeo.2025.102228","url":null,"abstract":"<div><div>We present a generic formulation of an algorithmic paradigm for approximating maximum weighted independent sets (MWIS) in arbitrary vertex weighted graphs. A special case of this paradigm has been proposed earlier for geometric intersection graphs. Here, we propose and analyse a much more general formulation. As part of this formulation, we introduce a new graph parameter which plays a role in bounding the approximation factor of the algorithms. By applying this paradigm to intersection graph classes of specific types of geometric objects, we obtain efficient algorithms which approximate a MWIS within <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> multiplicative factors. It is also shown that the same approach can be generalised to obtain efficient approximation algorithms for computing an optimal weight <span><math><mi>P</mi></math></span>-subgraphs where <span><math><mi>P</mi></math></span> is a suitable hereditary property.</div><div>Applying our paradigm, we establish, for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>]</mo></math></span>, that MWIS of the intersection graph of a given collection of weighted <em>k</em>-dimensional <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> spheres (having a common radius) can be efficiently approximated within a multiplicative factor of <span><math><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. The running time can be brought down to <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span> at the cost of increasing the approximation guarantee to <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>p</mi></mrow></msub><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, for some constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> depending only on <em>p</em> and <em>k</em>. It is also shown that the above MWIS-approximation results can be extended to MWIS-approximation over the more general intersection graphs of finite collections of connected, full-dimensional and centrally-symmetric bodies in <em>k</em>-dimensional, <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-spaces, for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>]</mo></math></span>.</div><div>In a related development, we also establish the following graph theoretic result which will be of independent interest: For every <span><math><mi>p</mi><mo>∈</","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102228"},"PeriodicalIF":0.7,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099437","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}
Pub Date : 2025-09-11DOI: 10.1016/j.comgeo.2025.102227
Gabriel Currier , Jozsef Solymosi , Hung-Hsun Hans Yu
In this note, we show that extremal Szemerédi–Trotter configurations are rigid in the following sense: If are sets of points and lines determining at least incidences, then there exists a collection of points of size at most such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth–Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester–Gallai configurations.
{"title":"On the structure of extremal point-line arrangements","authors":"Gabriel Currier , Jozsef Solymosi , Hung-Hsun Hans Yu","doi":"10.1016/j.comgeo.2025.102227","DOIUrl":"10.1016/j.comgeo.2025.102227","url":null,"abstract":"<div><div>In this note, we show that extremal Szemerédi–Trotter configurations are rigid in the following sense: If <span><math><mi>P</mi><mo>,</mo><mi>L</mi></math></span> are sets of points and lines determining at least <span><math><mi>C</mi><mo>|</mo><mi>P</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>|</mo><mi>L</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></math></span> incidences, then there exists a collection <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of points of size at most <span><math><mi>k</mi><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth–Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester–Gallai configurations.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102227"},"PeriodicalIF":0.7,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145099438","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}
Pub Date : 2025-09-11DOI: 10.1016/j.comgeo.2025.102219
Helmut Alt , Sergio Cabello , Otfried Cheong , Ji-won Park , Nadja Seiferth
In this article, we consider the problems of finding in dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of d-dimensional unit-radius balls can be packed under translations. The computational problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm for each of these containers based on a reduction to finding a shortest Hamiltonian path in a weighted graph, which in turn models the problem of stabbing the centers of the input balls while keeping them disjoint. We also show that for n such balls, a container of volume is always sufficient and sometimes necessary. As a byproduct, this implies that for there is no finite size -dimensional convex body into which all d-dimensional unit-radius balls can be packed simultaneously.
{"title":"Packing d-dimensional balls into a d + 1-dimensional container","authors":"Helmut Alt , Sergio Cabello , Otfried Cheong , Ji-won Park , Nadja Seiferth","doi":"10.1016/j.comgeo.2025.102219","DOIUrl":"10.1016/j.comgeo.2025.102219","url":null,"abstract":"<div><div>In this article, we consider the problems of finding in <span><math><mi>d</mi><mo>+</mo><mn>1</mn></math></span> dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of <em>d</em>-dimensional unit-radius balls can be packed under translations. The computational problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm for each of these containers based on a reduction to finding a shortest Hamiltonian path in a weighted graph, which in turn models the problem of stabbing the centers of the input balls while keeping them disjoint. We also show that for <em>n</em> such balls, a container of volume <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></mfrac></mrow></msup><mo>)</mo></math></span> is always sufficient and sometimes necessary. As a byproduct, this implies that for <span><math><mi>d</mi><mo>⩾</mo><mn>2</mn></math></span> there is no finite size <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-dimensional convex body into which all <em>d</em>-dimensional unit-radius balls can be packed simultaneously.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"132 ","pages":"Article 102219"},"PeriodicalIF":0.7,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145120133","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}
We study the problem of multiple anchored watchman routes, where we are given m starting points for watchmen, and aim to find routes for all watchmen such that all points in a polygon are visible from at least one route. We consider the problem in Minbar polygons,2 which are staircase polygons for which the floor of the staircase solely consists of one horizontal and one vertical edge, and in generalized Minbar polygons, which relaxes the definition of Minbar polygons, allowing for non-rectilinear edges. For Minbar polygons, we exhibit polynomial time algorithms to compute optimal solutions for both the min-max and the min-sum criteria. The min-max algorithm takes time, using storage, and the min-sum algorithm takes time, also using storage.
For generalized Minbar polygons, we prove NP-hardness for the min-sum and min-max criteria, and present approximation algorithms for both criteria: an -approximation taking time for the min-sum criterion, and a -approximation taking time for the min-max criterion.
Minbar polygons and the non-rectilinear generalization of them may seem to be very restricted polygon classes but they form an adjacent pair where the multiple anchored watchman routes problem has a polynomial time solution in one class but is NP-hard in the slightly more generalized class. It is this property that motivates our study of these restricted polygon classes.
{"title":"m-Watchmen's routes in minbar and generalized minbar polygons","authors":"Rahmat Ghasemi , Alireza Bagheri , Anna Brötzner , Fatemeh Keshavarz-Kohjerdi , Faezeh Farivar , Bengt J. Nilsson , Christiane Schmidt","doi":"10.1016/j.comgeo.2025.102217","DOIUrl":"10.1016/j.comgeo.2025.102217","url":null,"abstract":"<div><div>We study the problem of multiple anchored watchman routes, where we are given <em>m</em> starting points for watchmen, and aim to find routes for all watchmen such that all points in a polygon are visible from at least one route. We consider the problem in Minbar polygons,<span><span><sup>2</sup></span></span> which are staircase polygons for which the floor of the staircase solely consists of one horizontal and one vertical edge, and in generalized Minbar polygons, which relaxes the definition of Minbar polygons, allowing for non-rectilinear edges. For Minbar polygons, we exhibit polynomial time algorithms to compute optimal solutions for both the min-max and the min-sum criteria. The min-max algorithm takes <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>log</mi><mo></mo><mi>m</mi><mo>+</mo><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time, using <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span> storage, and the min-sum algorithm takes <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mi>m</mi><mo>+</mo><mi>m</mi><mi>log</mi><mo></mo><mi>m</mi><mo>)</mo></math></span> time, also using <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span> storage.</div><div>For generalized Minbar polygons, we prove NP-hardness for the min-sum and min-max criteria, and present approximation algorithms for both criteria: an <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo></mo><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>-approximation taking <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>4</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time for the min-sum criterion, and a <span><math><mo>(</mo><mi>π</mi><mo>+</mo><mn>3</mn><mo>)</mo></math></span>-approximation taking <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time for the min-max criterion.</div><div>Minbar polygons and the non-rectilinear generalization of them may seem to be very restricted polygon classes but they form an adjacent pair where the multiple anchored watchman routes problem has a polynomial time solution in one class but is NP-hard in the slightly more generalized class. It is this property that motivates our study of these restricted polygon classes.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102217"},"PeriodicalIF":0.7,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144932142","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}
Pub Date : 2025-08-26DOI: 10.1016/j.comgeo.2025.102218
David Conlon , Jacob Fox , Xiaoyu He , Dhruv Mubayi , Andrew Suk , Jacques Verstraëte
Let be the minimum N such that every N-element point set in the plane contains either ℓ collinear members or n points in convex position. We prove that there is a constant such that, for each , A similar extension of the well-known Erdős–Szekeres cups-caps theorem is also proved.
设ES (n)为最小n，使得平面上的每个n元素点集包含l个共线成员或n个凸位置上的点。我们证明了存在一个常数C>；0，使得对于每一个n≥3的，（3r−1）·2n−5<ES r (n)< r 2·2n+Cnlog ln n。我们还证明了著名的Erdős-Szekeres杯帽定理的一个类似推广。
{"title":"Big line or big convex polygon","authors":"David Conlon , Jacob Fox , Xiaoyu He , Dhruv Mubayi , Andrew Suk , Jacques Verstraëte","doi":"10.1016/j.comgeo.2025.102218","DOIUrl":"10.1016/j.comgeo.2025.102218","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> be the minimum <em>N</em> such that every <em>N</em>-element point set in the plane contains either <em>ℓ</em> collinear members or <em>n</em> points in convex position. We prove that there is a constant <span><math><mi>C</mi><mo>></mo><mn>0</mn></math></span> such that, for each <span><math><mi>ℓ</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>,<span><span><span><math><mo>(</mo><mn>3</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>5</mn></mrow></msup><mo><</mo><mi>E</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo><</mo><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>+</mo><mi>C</mi><msqrt><mrow><mi>n</mi><mi>log</mi><mo></mo><mi>n</mi></mrow></msqrt></mrow></msup><mo>.</mo></math></span></span></span> A similar extension of the well-known Erdős–Szekeres cups-caps theorem is also proved.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102218"},"PeriodicalIF":0.7,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144913031","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}
Pub Date : 2025-06-26DOI: 10.1016/j.comgeo.2025.102210
Byeonguk Kang , Junhyeok Choi , Jeesun Han , Hee-Kap Ahn
We study the problem of guarding points on an x-monotone polygonal chain, called a terrain, using k watchtowers. A watchtower is a vertical segment whose bottom endpoint lies on the terrain. A point on the terrain is visible from a watchtower if the line segment connecting the point and the top endpoint of the watchtower does not cross the terrain. Given a sequence of point sites lying on a terrain, we aim to partition the sequence into k contiguous subsequences and place k watchtowers on the terrain such that every point site in a subsequence is visible from the same watchtower and the maximum length of the watchtowers is minimized. We present efficient algorithms for two variants of the problem.
{"title":"Guarding points on a terrain by watchtowers","authors":"Byeonguk Kang , Junhyeok Choi , Jeesun Han , Hee-Kap Ahn","doi":"10.1016/j.comgeo.2025.102210","DOIUrl":"10.1016/j.comgeo.2025.102210","url":null,"abstract":"<div><div>We study the problem of guarding points on an <em>x</em>-monotone polygonal chain, called a terrain, using <em>k</em> watchtowers. A watchtower is a vertical segment whose bottom endpoint lies on the terrain. A point on the terrain is visible from a watchtower if the line segment connecting the point and the top endpoint of the watchtower does not cross the terrain. Given a sequence of point sites lying on a terrain, we aim to partition the sequence into <em>k</em> contiguous subsequences and place <em>k</em> watchtowers on the terrain such that every point site in a subsequence is visible from the same watchtower and the maximum length of the watchtowers is minimized. We present efficient algorithms for two variants of the problem.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"131 ","pages":"Article 102210"},"PeriodicalIF":0.4,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144523131","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}
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