Discrete & Computational Geometry最新文献

{"title":"Maximum Matchings in Geometric Intersection Graphs.","authors":"Édouard Bonnet,&nbsp;Sergio Cabello,&nbsp;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}

{"title":"Nearly <i>k</i>-Distance Sets.","authors":"Nóra Frankl,&nbsp;Andrey Kupavskii","doi":"10.1007/s00454-023-00489-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00489-x","url":null,"abstract":"&lt;p&gt;&lt;p&gt;We say that a set of points &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt; is an &lt;math&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/math&gt;-nearly &lt;i&gt;k&lt;/i&gt;-distance set if there exist &lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;, such that the distance between any two distinct points in &lt;i&gt;S&lt;/i&gt; falls into &lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;. In this paper, we study the quantity &lt;dispformula&gt;&lt;math&gt;&lt;mrow&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;lim&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;max&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mtext&gt;is an&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mtext&gt;-nearly&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mtext&gt;-distance set in&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/msup&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/dispformula&gt;and its relation to the classical quantity &lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;: the size of the largest &lt;i&gt;k&lt;/i&gt;-distance set in &lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;. We obtain that &lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt; for &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;, as well as for any fixed &lt;i&gt;k&lt;/i&gt;, provided that &lt;i&gt;d&lt;/i&gt; 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 &lt;i&gt;n&lt;/i&gt; points in &lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;, how many pairs of them form a distance that belongs to &lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;∪&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;, where &lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt; 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 &lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;-&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;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}

{"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}

{"title":"Cusp Density and Commensurability of Non-arithmetic Hyperbolic Coxeter Orbifolds.","authors":"Edoardo Dotti,&nbsp;Simon T Drewitz,&nbsp;Ruth Kellerhals","doi":"10.1007/s00454-022-00455-z","DOIUrl":"https://doi.org/10.1007/s00454-022-00455-z","url":null,"abstract":"<p><p>For three distinct infinite families <math><mrow><mo>(</mo> <msub><mi>R</mi> <mi>m</mi></msub> <mo>)</mo></mrow> </math> , <math><mrow><mo>(</mo> <msub><mi>S</mi> <mi>m</mi></msub> <mo>)</mo></mrow> </math> , and <math><mrow><mo>(</mo> <msub><mi>T</mi> <mi>m</mi></msub> <mo>)</mo></mrow> </math> of non-arithmetic 1-cusped hyperbolic Coxeter 3-orbifolds, we prove incommensurability for a pair of elements <math><msub><mi>X</mi> <mi>k</mi></msub> </math> and <math><msub><mi>Y</mi> <mi>l</mi></msub> </math> belonging to the same sequence and for most pairs belonging two different ones. We investigate this problem first by means of the Vinberg space and the Vinberg form, a quadratic space associated to each of the corresponding fundamental Coxeter prism groups, which allows us to deduce some partial results. The complete proof is based on the analytic behavior of another commensurability invariant. It is given by the cusp density, and we prove and exploit its strict monotonicity.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 3","pages":"873-895"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9984359/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9424995","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}

{"title":"Deletion in Abstract Voronoi Diagrams in Expected Linear Time and Related Problems.","authors":"Kolja Junginger,&nbsp;Evanthia Papadopoulou","doi":"10.1007/s00454-022-00463-z","DOIUrl":"10.1007/s00454-022-00463-z","url":null,"abstract":"<p><p>Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem in a long time; similarly, for any concrete Voronoi diagram of generalized (non-point) sites. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion of one site. To achieve this result, we use the concept of a Voronoi-like diagram, a relaxed Voronoi structure of independent interest. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under insertion, therefore, enabling its use in incremental constructions. The time-complexity analysis introduces a variant to backwards analysis, which is applicable to order-dependent structures. We further extend the technique to compute in expected linear time: the order-<math><mrow><mo>(</mo><mi>k</mi><mspace></mspace><mo>+</mo><mspace></mspace><mn>1</mn><mo>)</mo></mrow></math> subdivision within an order-<i>k</i> Voronoi region, and the farthest abstract Voronoi diagram, after the order of its regions at infinity is known.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 4","pages":"1040-1078"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10169906/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10296942","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}

{"title":"Inserting One Edge into a Simple Drawing is Hard.","authors":"Alan Arroyo,&nbsp;Fabian Klute,&nbsp;Irene Parada,&nbsp;Birgit Vogtenhuber,&nbsp;Raimund Seidel,&nbsp;Tilo Wiedera","doi":"10.1007/s00454-022-00394-9","DOIUrl":"https://doi.org/10.1007/s00454-022-00394-9","url":null,"abstract":"<p><p>A <i>simple drawing</i> <i>D</i>(<i>G</i>) of a graph <i>G</i> is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge <i>e</i> in the complement of <i>G</i> can be <i>inserted</i> into <i>D</i>(<i>G</i>) if there exists a simple drawing of <math><mrow><mi>G</mi> <mo>+</mo> <mi>e</mi></mrow> </math> extending <i>D</i>(<i>G</i>). As a result of Levi's Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of <i>G</i> can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles <math><mi>A</mi></math> and a pseudosegment  <math><mi>σ</mi></math> , it can be decided in polynomial time whether there exists a pseudocircle <math><msub><mi>Φ</mi> <mi>σ</mi></msub> </math> extending <math><mi>σ</mi></math> for which <math><mrow><mi>A</mi> <mo>∪</mo> <mo>{</mo> <msub><mi>Φ</mi> <mi>σ</mi></msub> <mo>}</mo></mrow> </math> is again an arrangement of pseudocircles.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 3","pages":"745-770"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9984358/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9424996","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}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Completeness for the Complexity Class <ns0:math><ns0:mrow><ns0:mo>∀</ns0:mo><ns0:mo>∃</ns0:mo><ns0:mi>R</ns0:mi></ns0:mrow></ns0:math> and Area-Universality.","authors":"Michael Gene Dobbins,&nbsp;Linda Kleist,&nbsp;Tillmann Miltzow,&nbsp;Paweł Rzążewski","doi":"10.1007/s00454-022-00381-0","DOIUrl":"10.1007/s00454-022-00381-0","url":null,"abstract":"<p><p>Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class <math><mrow><mo>∃</mo><mi>R</mi></mrow></math> plays a crucial role in the study of geometric problems. Sometimes <math><mrow><mo>∃</mo><mi>R</mi></mrow></math> is referred to as the 'real analog' of NP. While NP is a class of computational problems that deals with existentially quantified <i>boolean</i> variables, <math><mrow><mo>∃</mo><mi>R</mi></mrow></math> deals with existentially quantified <i>real</i> variables. In analogy to <math><msubsup><mi>Π</mi><mn>2</mn><mi>p</mi></msubsup></math> and <math><msubsup><mi>Σ</mi><mn>2</mn><mi>p</mi></msubsup></math> in the famous polynomial hierarchy, we study the complexity classes <math><mrow><mo>∀</mo><mo>∃</mo><mi>R</mi></mrow></math> and <math><mrow><mo>∃</mo><mo>∀</mo><mi>R</mi></mrow></math> with <i>real</i> variables. Our main interest is the Area Universality problem, where we are given a plane graph <i>G</i>, and ask if for each assignment of areas to the inner faces of <i>G</i>, there exists a straight-line drawing of <i>G</i> realizing the assigned areas. We conjecture that Area Universality is <math><mrow><mo>∀</mo><mo>∃</mo><mi>R</mi></mrow></math>-complete and support this conjecture by proving <math><mrow><mo>∃</mo><mi>R</mi></mrow></math>- and <math><mrow><mo>∀</mo><mo>∃</mo><mi>R</mi></mrow></math>-completeness of two variants of Area Universality. To this end, we introduce tools to prove <math><mrow><mo>∀</mo><mo>∃</mo><mi>R</mi></mrow></math>-hardness and membership. Finally, we present geometric problems as candidates for <math><mrow><mo>∀</mo><mo>∃</mo><mi>R</mi></mrow></math>-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 1","pages":"154-188"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10244296/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9600927","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}

{"title":"Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile.","authors":"Rachel Greenfeld, Terence Tao","doi":"10.1007/s00454-022-00426-4","DOIUrl":"10.1007/s00454-022-00426-4","url":null,"abstract":"<p><p>We construct an example of a group <math><mrow><mi>G</mi><mo>=</mo><msup><mrow><mi>Z</mi></mrow><mn>2</mn></msup><mo>×</mo><msub><mi>G</mi><mn>0</mn></msub></mrow></math> for a finite abelian group <math><msub><mi>G</mi><mn>0</mn></msub></math>, a subset <i>E</i> of <math><msub><mi>G</mi><mn>0</mn></msub></math>, and two finite subsets <math><mrow><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub></mrow></math> of <i>G</i>, such that it is undecidable in ZFC whether <math><mrow><msup><mrow><mi>Z</mi></mrow><mn>2</mn></msup><mo>×</mo><mi>E</mi></mrow></math> can be tiled by translations of <math><mrow><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub></mrow></math>. In particular, this implies that this tiling problem is <i>aperiodic</i>, in the sense that (in the standard universe of ZFC) there exist translational tilings of <i>E</i> by the tiles <math><mrow><msub><mi>F</mi><mn>1</mn></msub><mo>,</mo><msub><mi>F</mi><mn>2</mn></msub></mrow></math>, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in <math><msup><mrow><mi>Z</mi></mrow><mn>2</mn></msup></math>). A similar construction also applies for <math><mrow><mi>G</mi><mo>=</mo><msup><mrow><mi>Z</mi></mrow><mi>d</mi></msup></mrow></math> for sufficiently large <i>d</i>. If one allows the group <math><msub><mi>G</mi><mn>0</mn></msub></math> to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile <i>F</i>. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"1 1","pages":"1652-1706"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10676348/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49166205","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}

{"title":"Local Criteria for Triangulating General Manifolds.","authors":"Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Mathijs Wintraecken","doi":"10.1007/s00454-022-00431-7","DOIUrl":"10.1007/s00454-022-00431-7","url":null,"abstract":"<p><p>We present criteria for establishing a triangulation of a manifold. Given a manifold <i>M</i>, a simplicial complex  <math><mi>A</mi></math> , and a map <i>H</i> from the underlying space of <math><mi>A</mi></math> to <i>M</i>, our criteria are presented in local coordinate charts for <i>M</i>, and ensure that <i>H</i> is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on <i>M</i>. No Delaunay property of <math><mi>A</mi></math> is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 1","pages":"156-191"},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9805998/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10860065","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}

{"title":"Computing the Multicover Bifiltration.","authors":"René Corbet,&nbsp;Michael Kerber,&nbsp;Michael Lesnick,&nbsp;Georg Osang","doi":"10.1007/s00454-022-00476-8","DOIUrl":"10.1007/s00454-022-00476-8","url":null,"abstract":"<p><p>Given a finite set <math><mrow><mi>A</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mi>d</mi></msup></mrow></math>, let <math><msub><mtext>Cov</mtext><mrow><mi>r</mi><mo>,</mo><mi>k</mi></mrow></msub></math> denote the set of all points within distance <i>r</i> to at least <i>k</i> points of <i>A</i>. Allowing <i>r</i> and <i>k</i> to vary, we obtain a 2-parameter family of spaces that grow larger when <i>r</i> increases or <i>k</i> decreases, called the <i>multicover bifiltration</i>. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the <i>rhomboid tiling</i> of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"70 2","pages":"376-405"},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10423148/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10005973","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}