Discrete and Computational Geometry最新文献

{"title":"Deformed Graphical Zonotopes","authors":"Arnau Padrol, Vincent Pilaud, Germain Poullot","doi":"10.1007/s00454-023-00586-x","DOIUrl":"https://doi.org/10.1007/s00454-023-00586-x","url":null,"abstract":"Abstract We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph G , consisting of independent equations defining its linear span (in terms of non-cliques of G ) and of the inequalities defining its facets (in terms of common neighbors of neighbors in G ). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in G form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if G is triangle-free.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"1702 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135805176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extracting Persistent Clusters in Dynamic Data via Möbius Inversion","authors":"Woojin Kim, Facundo Mémoli","doi":"10.1007/s00454-023-00590-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00590-1","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136098310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The VC-Dimension and Point Configurations in $${mathbb F}_q^2$$","authors":"David Fitzpatrick, Alex Iosevich, Brian McDonald, Emmett Wyman","doi":"10.1007/s00454-023-00570-5","DOIUrl":"https://doi.org/10.1007/s00454-023-00570-5","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136353542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Periodic Steiner Networks Minimizing Length","authors":"Jerome Alex, Karsten Grosse-Brauckmann","doi":"10.1007/s00454-023-00576-z","DOIUrl":"https://doi.org/10.1007/s00454-023-00576-z","url":null,"abstract":"Abstract We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain with the same volume. The minimizer is the so-called network with quotient the complete graph on four vertices $$K_4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> . For comparison we consider a competing topological class, also with a quotient on four vertices, and determine the minimizing networks.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135043441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximating Maximum Integral Multiflows on Bounded Genus Graphs","authors":"Chien-chung Huang, Mathieu Mari, Claire Mathieu, Jens Vygen","doi":"10.1007/s00454-023-00552-7","DOIUrl":"https://doi.org/10.1007/s00454-023-00552-7","url":null,"abstract":"Abstract We devise the first constant-factor approximation algorithm for finding an integral multi-commodity flow of maximum total value for instances where the supply graph together with the demand edges can be embedded on an orientable surface of bounded genus. This extends recent results for planar instances. Our techniques include an uncrossing algorithm, which is significantly more difficult than in the planar case, a partition of the cycles in the support of an LP solution into free homotopy classes, and a new rounding procedure for freely homotopic non-separating cycles.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135043895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Ultrametric Gromov–Wasserstein Distance","authors":"Facundo Mémoli, Axel Munk, Zhengchao Wan, Christoph Weitkamp","doi":"10.1007/s00454-023-00583-0","DOIUrl":"https://doi.org/10.1007/s00454-023-00583-0","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135252209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications","authors":"Tamal K. Dey, Woojin Kim, Facundo Mémoli","doi":"10.1007/s00454-023-00584-z","DOIUrl":"https://doi.org/10.1007/s00454-023-00584-z","url":null,"abstract":"Abstract The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval I of a $$textbf{Z}^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank of M over I by computing the barcode of the zigzag module obtained by restricting to that path. If M is the homology of a bifiltration F of $$t$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> simplices (while accounting for multi-criticality) and I consists of $$t$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> points, this computation takes $$O(t^omega )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>ω</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> time where $$omega in [2,2.373)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2.373</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is the exponent of matrix multiplication. We apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M , determine whether M is interval decomposable and, if so, compute all intervals supporting its indecomposable summands.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"609 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135252056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Universal Triangulation for Flat Tori","authors":"Francis Lazarus, Florent Tallerie","doi":"10.1007/s00454-023-00598-7","DOIUrl":"https://doi.org/10.1007/s00454-023-00598-7","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"298 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Worst-Case Optimal Covering of Rectangles by Disks","authors":"Sándor P. Fekete, Utkarsh Gupta, Phillip Keldenich, Christian Scheffer, Sahil Shah","doi":"10.1007/s00454-023-00582-1","DOIUrl":"https://doi.org/10.1007/s00454-023-00582-1","url":null,"abstract":"Abstract We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any $$lambda ge 1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the critical covering area $$A^*(lambda )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is the minimum value for which any set of disks with total area at least $$A^*(lambda )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> can cover a rectangle of dimensions $$lambda times 1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>×</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We show that there is a threshold value $$lambda _2 = sqrt{sqrt{7}/2 - 1/4} approx 1.035797ldots $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mrow> <mml:msqrt> <mml:mn>7</mml:mn> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>≈</mml:mo> <mml:mn>1.035797</mml:mn> <mml:mo>…</mml:mo> </mml:mrow> </mml:math> , such that for $$lambda <lambda _2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo><</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> the critical covering area $$A^*(lambda )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is $$A^*(lambda )=3pi left( frac{lambda ^2}{16} +frac{5}{32} + frac{9}{256lambda ^2}right) $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mi>π</mml:mi> <mml:mfenced> <mml:mfrac> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mn>16</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:mn>32</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>9</mml:mn> <mml:mrow> <mml:mn>256</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mfenced> </mml:mrow> </mml:math> , and for $$lambda","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Posets Arising as 1-Skeleta of Simple Polytopes, the Nonrevisiting Path Conjecture, and Poset Topology","authors":"Patricia Hersh","doi":"10.1007/s00454-023-00588-9","DOIUrl":"https://doi.org/10.1007/s00454-023-00588-9","url":null,"abstract":"","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}