Pub Date : 2023-05-16DOI: 10.1007/s00454-023-00485-1
D. Cohen-Steiner, A. Lieutier, J. Vuillamy
{"title":"Delaunay and Regular Triangulations as Lexicographic Optimal Chains","authors":"D. Cohen-Steiner, A. Lieutier, J. Vuillamy","doi":"10.1007/s00454-023-00485-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00485-1","url":null,"abstract":"","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 1","pages":"1 - 50"},"PeriodicalIF":0.8,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45486294","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}
Pub Date : 2023-01-01DOI: 10.1007/s00454-022-00446-0
Georg Loho, Raman Sanyal
Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.
{"title":"Tropical Carathéodory with Matroids.","authors":"Georg Loho, Raman Sanyal","doi":"10.1007/s00454-022-00446-0","DOIUrl":"https://doi.org/10.1007/s00454-022-00446-0","url":null,"abstract":"<p><p>Bárány's colorful generalization of Carathéodory's Theorem combines geometrical and combinatorial constraints. Kalai-Meshulam (2005) and Holmsen (2016) generalized Bárány's theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert-Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 1","pages":"139-155"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9805987/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10494381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01Epub Date: 2023-03-22DOI: 10.1007/s00454-022-00414-8
Katrin Casel, Henning Fernau, Alexander Grigoriev, Markus L Schmid, Sue Whitesides
Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is -hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be -hard, which settles an open question.
{"title":"Combinatorial Properties and Recognition of Unit Square Visibility Graphs.","authors":"Katrin Casel, Henning Fernau, Alexander Grigoriev, Markus L Schmid, Sue Whitesides","doi":"10.1007/s00454-022-00414-8","DOIUrl":"10.1007/s00454-022-00414-8","url":null,"abstract":"<p><p>Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is <math><mrow><mspace></mspace><mrow><mi>N</mi><mi>P</mi></mrow><mspace></mspace></mrow></math>-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be <math><mrow><mspace></mspace><mrow><mi>N</mi><mi>P</mi></mrow><mspace></mspace></mrow></math>-hard, which settles an open question.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 4","pages":"937-980"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10169907/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10301082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01Epub Date: 2023-09-09DOI: 10.1007/s00454-023-00564-3
Édouard Bonnet, Sergio Cabello, Wolfgang Mulzer
Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in time with high probability, where is the density of the geometric objects and is a constant such that matrices can be multiplied in time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in can be found in time with high probability.
{"title":"Maximum Matchings in Geometric Intersection Graphs.","authors":"Édouard Bonnet, Sergio Cabello, Wolfgang Mulzer","doi":"10.1007/s00454-023-00564-3","DOIUrl":"10.1007/s00454-023-00564-3","url":null,"abstract":"<p><p>Let <i>G</i> be an intersection graph of <i>n</i> geometric objects in the plane. We show that a maximum matching in <i>G</i> can be found in <math><mrow><mi>O</mi><mspace></mspace><mo>(</mo><msup><mi>ρ</mi><mrow><mn>3</mn><mi>ω</mi><mo>/</mo><mn>2</mn></mrow></msup><msup><mi>n</mi><mrow><mi>ω</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></math> time with high probability, where <math><mi>ρ</mi></math> is the density of the geometric objects and <math><mrow><mi>ω</mi><mo>></mo><mn>2</mn></mrow></math> is a constant such that <math><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></math> matrices can be multiplied in <math><mrow><mi>O</mi><mo>(</mo><msup><mi>n</mi><mi>ω</mi></msup><mo>)</mo></mrow></math> time. The same result holds for any subgraph of <i>G</i>, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in <math><mrow><mi>O</mi><mo>(</mo><msup><mi>n</mi><mrow><mi>ω</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></math> time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in <math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>Ψ</mi><mo>]</mo></mrow></math> can be found in <math><mrow><mi>O</mi><mspace></mspace><mo>(</mo><msup><mi>Ψ</mi><mn>6</mn></msup><msup><mo>log</mo><mn>11</mn></msup><mspace></mspace><mi>n</mi><mo>+</mo><msup><mi>Ψ</mi><mrow><mn>12</mn><mi>ω</mi></mrow></msup><msup><mi>n</mi><mrow><mi>ω</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></math> time with high probability.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 3","pages":"550-579"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10550895/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41156223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01Epub Date: 2023-06-06DOI: 10.1007/s00454-023-00489-x
Nóra Frankl, Andrey Kupavskii
<p><p>We say that a set of points <math><mrow><mi>S</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup></mrow></math> is an <math><mi>ε</mi></math>-nearly <i>k</i>-distance set if there exist <math><mrow><mn>1</mn><mo>≤</mo><msub><mi>t</mi><mn>1</mn></msub><mo>≤</mo><mo>…</mo><mo>≤</mo><msub><mi>t</mi><mi>k</mi></msub></mrow></math>, such that the distance between any two distinct points in <i>S</i> falls into <math><mrow><mrow><mo>[</mo><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><mi>ε</mi><mo>]</mo></mrow><mo>∪</mo><mo>⋯</mo><mo>∪</mo><mrow><mo>[</mo><msub><mi>t</mi><mi>k</mi></msub><mo>,</mo><msub><mi>t</mi><mi>k</mi></msub><mo>+</mo><mi>ε</mi><mo>]</mo></mrow></mrow></math>. In this paper, we study the quantity <dispformula><math><mrow><mtable><mtr><mtd><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>=</mo><munder><mo>lim</mo><mrow><mi>ε</mi><mo>→</mo><mn>0</mn></mrow></munder><mo>max</mo><mrow><mo>{</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>:</mo><mi>S</mi><mspace></mspace><mspace></mspace><mtext>is an</mtext><mspace></mspace><mi>ε</mi><mtext>-nearly</mtext><mspace></mspace><mi>k</mi><mtext>-distance set in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup><mo>}</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>and its relation to the classical quantity <math><mrow><msub><mi>m</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></math>: the size of the largest <i>k</i>-distance set in <math><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup></math>. We obtain that <math><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>m</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>, as well as for any fixed <i>k</i>, provided that <i>d</i> is sufficiently large. The last result answers a question, proposed by Erdős, Makai, and Pach. We also address a closely related Turán-type problem, studied by Erdős, Makai, Pach, and Spencer in the 90s: given <i>n</i> points in <math><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup></math>, how many pairs of them form a distance that belongs to <math><mrow><mrow><mo>[</mo><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo>]</mo></mrow><mo>∪</mo><mo>⋯</mo><mo>∪</mo><mrow><mo>[</mo><msub><mi>t</mi><mi>k</mi></msub><mo>,</mo><msub><mi>t</mi><mi>k</mi></msub><mo>+</mo><mn>1</mn><mo>]</mo></mrow></mrow></math>, where <math><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>t</mi><mi>k</mi></msub></mrow></math> are fixed and any two points in the set are at distance at least 1 apart? We establish the connection between this quantity and a quantity closely related to <math><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>-</mo><mn>1</mn><m
{"title":"Nearly <i>k</i>-Distance Sets.","authors":"Nóra Frankl, Andrey Kupavskii","doi":"10.1007/s00454-023-00489-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00489-x","url":null,"abstract":"<p><p>We say that a set of points <math><mrow><mi>S</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup></mrow></math> is an <math><mi>ε</mi></math>-nearly <i>k</i>-distance set if there exist <math><mrow><mn>1</mn><mo>≤</mo><msub><mi>t</mi><mn>1</mn></msub><mo>≤</mo><mo>…</mo><mo>≤</mo><msub><mi>t</mi><mi>k</mi></msub></mrow></math>, such that the distance between any two distinct points in <i>S</i> falls into <math><mrow><mrow><mo>[</mo><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><mi>ε</mi><mo>]</mo></mrow><mo>∪</mo><mo>⋯</mo><mo>∪</mo><mrow><mo>[</mo><msub><mi>t</mi><mi>k</mi></msub><mo>,</mo><msub><mi>t</mi><mi>k</mi></msub><mo>+</mo><mi>ε</mi><mo>]</mo></mrow></mrow></math>. In this paper, we study the quantity <dispformula><math><mrow><mtable><mtr><mtd><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>=</mo><munder><mo>lim</mo><mrow><mi>ε</mi><mo>→</mo><mn>0</mn></mrow></munder><mo>max</mo><mrow><mo>{</mo><mo>|</mo><mi>S</mi><mo>|</mo><mo>:</mo><mi>S</mi><mspace></mspace><mspace></mspace><mtext>is an</mtext><mspace></mspace><mi>ε</mi><mtext>-nearly</mtext><mspace></mspace><mi>k</mi><mtext>-distance set in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup><mo>}</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></dispformula>and its relation to the classical quantity <math><mrow><msub><mi>m</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></math>: the size of the largest <i>k</i>-distance set in <math><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup></math>. We obtain that <math><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow><mo>=</mo><msub><mi>m</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>, as well as for any fixed <i>k</i>, provided that <i>d</i> is sufficiently large. The last result answers a question, proposed by Erdős, Makai, and Pach. We also address a closely related Turán-type problem, studied by Erdős, Makai, Pach, and Spencer in the 90s: given <i>n</i> points in <math><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup></math>, how many pairs of them form a distance that belongs to <math><mrow><mrow><mo>[</mo><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo>]</mo></mrow><mo>∪</mo><mo>⋯</mo><mo>∪</mo><mrow><mo>[</mo><msub><mi>t</mi><mi>k</mi></msub><mo>,</mo><msub><mi>t</mi><mi>k</mi></msub><mo>+</mo><mn>1</mn><mo>]</mo></mrow></mrow></math>, where <math><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>t</mi><mi>k</mi></msub></mrow></math> are fixed and any two points in the set are at distance at least 1 apart? We establish the connection between this quantity and a quantity closely related to <math><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>d</mi><mo>-</mo><mn>1</mn><m","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 3","pages":"455-494"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10550902/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41158582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01Epub Date: 2022-09-29DOI: 10.1007/s00454-022-00428-2
Maximilian Jaroschek, Manuel Kauers, Laura Kovács
Given a lattice and a subset , we say that a point in A is lonely if it is not equivalent modulo to another point of A. We are interested in identifying lonely points for specific choices of when A is a dilated standard simplex, and in conditions on which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.
给定一个网格 L ⊆ Z m 和一个子集 A ⊆ R m，如果 A 中的一个点不等价于 A 中的另一个点，那么我们就说这个点是孤点。我们感兴趣的是，当 A 是一个扩张的标准单纯形时，在 L 的特定选择下识别孤点，以及 L 的条件，这些条件可以确保孤点的数量在单纯形扩张到无穷大时是无限制的。
{"title":"Lonely Points in Simplices.","authors":"Maximilian Jaroschek, Manuel Kauers, Laura Kovács","doi":"10.1007/s00454-022-00428-2","DOIUrl":"10.1007/s00454-022-00428-2","url":null,"abstract":"<p><p>Given a lattice <math><mrow><mi>L</mi> <mo>⊆</mo> <msup><mi>Z</mi> <mi>m</mi></msup> </mrow> </math> and a subset <math><mrow><mi>A</mi> <mo>⊆</mo> <msup><mi>R</mi> <mi>m</mi></msup> </mrow> </math> , we say that a point in <i>A</i> is <i>lonely</i> if it is not equivalent modulo <math><mi>L</mi></math> to another point of <i>A</i>. We are interested in identifying lonely points for specific choices of <math><mi>L</mi></math> when <i>A</i> is a dilated standard simplex, and in conditions on <math><mi>L</mi></math> which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 1","pages":"4-25"},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9805990/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10481510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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