Pub Date : 2026-01-14DOI: 10.1016/j.comgeo.2026.102243
Sanjana Das , Adam Sheffer
We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances between points on two lines or curves. Unlike these existing problems, this new variant only makes sense if the points satisfy an additional spacing condition.
Our work can also be seen as an exploration of the proximity technique that was recently introduced by Solymosi and Zahl. This technique lies at the heart of our analysis.
{"title":"Distinct distances between a line and strip","authors":"Sanjana Das , Adam Sheffer","doi":"10.1016/j.comgeo.2026.102243","DOIUrl":"10.1016/j.comgeo.2026.102243","url":null,"abstract":"<div><div>We introduce a new type of distinct distances result: a lower bound on the number of distances between points on a line and points on a two-dimensional strip. This can be seen as a generalization of the well-studied problems of distances between points on two lines or curves. Unlike these existing problems, this new variant only makes sense if the points satisfy an additional spacing condition.</div><div>Our work can also be seen as an exploration of the proximity technique that was recently introduced by Solymosi and Zahl. This technique lies at the heart of our analysis.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102243"},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977137","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 : 2026-01-14DOI: 10.1016/j.comgeo.2026.102244
József Balogh , Haoran Luo
Let be the maximum possible size of a point set in general position in the p-random subset of . In this note, we determine the order of magnitude of up to a polylogarithmic factor by proving a balanced supersaturation result for the sets of 4 points in the same plane.
{"title":"Maximum number of points in general position in a random subset of finite 3-dimensional spaces","authors":"József Balogh , Haoran Luo","doi":"10.1016/j.comgeo.2026.102244","DOIUrl":"10.1016/j.comgeo.2026.102244","url":null,"abstract":"<div><div>Let <span><math><mi>α</mi><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>,</mo><mi>p</mi><mo>)</mo></math></span> be the maximum possible size of a point set in general position in the <em>p</em>-random subset of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>. In this note, we determine the order of magnitude of <span><math><mi>α</mi><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>,</mo><mi>p</mi><mo>)</mo></math></span> up to a polylogarithmic factor by proving a balanced supersaturation result for the sets of 4 points in the same plane.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102244"},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977136","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 : 2025-12-29DOI: 10.1016/j.comgeo.2025.102242
Atsuhiro Nakamoto , Kyosuke Wakayama
The rectangular drawing on the plane is a drawing of a plane graph which satisfies the following geometric conditions; each edge is represented as a horizontal or vertical line segment, and the boundary of each face has exactly four corners with angle . The necessary and sufficient condition for a subcubic plane graph to have a rectangular drawing was already given by Rahman et al. [14]. This result was extended to a drawing of graphs on an annulus by Hasheminezhad et al. [5]. However, in this paper, we point out an error in the result and get a necessary and sufficient condition for the graph to have a rectangular drawing on an annulus. By a simple analogy, we also obtain a similar result for the rectangular drawing on a Möbius band.
{"title":"Rectangular drawing of cubic graphs on an annulus and a Möbius band","authors":"Atsuhiro Nakamoto , Kyosuke Wakayama","doi":"10.1016/j.comgeo.2025.102242","DOIUrl":"10.1016/j.comgeo.2025.102242","url":null,"abstract":"<div><div>The rectangular drawing on the plane is a drawing of a plane graph which satisfies the following geometric conditions; each edge is represented as a horizontal or vertical line segment, and the boundary of each face has exactly four corners with angle <span><math><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. The necessary and sufficient condition for a subcubic plane graph to have a rectangular drawing was already given by Rahman et al. <span><span>[14]</span></span>. This result was extended to a drawing of graphs on an annulus by Hasheminezhad et al. <span><span>[5]</span></span>. However, in this paper, we point out an error in the result and get a necessary and sufficient condition for the graph to have a rectangular drawing on an annulus. By a simple analogy, we also obtain a similar result for the rectangular drawing on a Möbius band.</div></div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":"133 ","pages":"Article 102242"},"PeriodicalIF":0.7,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925040","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 : 2025-11-19DOI: 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}
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