Abstract In the study of parameterized streaming complexity on graph problems, the main goal is to design streaming algorithms for parameterized problems such that $$mathcal {O}(f(k) log ^{mathcal {O}(1)} n)$$ O(f(k)logO(1)n) space is enough, where f is an arbitrary computable function depending only on the parameter k . However, in the past few years very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size k of the solution we are seeking. Chitnis et al. (SODA’16) have shown that many important parameterized problems that form the backbone of traditional parameterized complexity are known to require $$Omega (n)$$ Ω(n) bits of storage for any streaming algorithm; e.g. Feedback Vertex Set , Even Cycle Transversal , Odd Cycle Transversal , Triangle Deletion or the more general $$mathcal{F}$$ F - Subgraph Deletion when parameterized by solution size k . Our contribution lies in overcoming the obstacles to efficient parameterized streaming algorithms in graph deletion problems by utilizing the power of parameterization. We focus on the vertex cover size K as the parameter for the parameterized graph deletion problems we consider. In this work, we consider the four most well-studied streaming models: the Ea , Dea , Va (vertex arrival) and Al (adjacency list) models. Surprisingly, the consideration of vertex cover size K in the different models leads to a classification of positive and negative results for problems like $$mathcal{F}$$ F - Subgraph Deletion and $$mathcal{F}$$ F - Minor Deletion .
在图问题的参数化流复杂性研究中，主要目标是设计参数化问题的流算法，使得$$mathcal {O}(f(k) log ^{mathcal {O}(1)} n)$$ O (f (k) log O (1) n)空间足够，其中f是一个仅依赖于参数k的任意可计算函数。然而，在过去几年中，取得的积极成果很少。大多数具有上述性质的流算法的图问题都是需要局部检查的问题，例如我们正在寻找的解决方案的大小k参数化的顶点覆盖或最大匹配。Chitnis等人(SODA ' 16)已经表明，许多重要的参数化问题构成了传统参数化复杂性的主干，已知任何流算法都需要$$Omega (n)$$ Ω (n)位存储;例如，反馈顶点集，偶环截线，奇环截线，三角形删除或更一般的$$mathcal{F}$$ F -子图删除，当解大小为k参数化时。我们的贡献在于利用参数化的力量克服了有效的参数化流算法在图删除问题中的障碍。我们关注顶点覆盖大小K作为我们考虑的参数化图删除问题的参数。在这项工作中，我们考虑了四种研究得最充分的流模型:Ea, Dea, Va(顶点到达)和Al(邻接表)模型。令人惊讶的是，不同模型中对顶点覆盖大小K的考虑导致了对$$mathcal{F}$$ F - Subgraph Deletion和$$mathcal{F}$$ F - Minor Deletion等问题的正面和负面结果的分类。
{"title":"Small Vertex Cover Helps in Fixed-Parameter Tractability of Graph Deletion Problems over Data Streams","authors":"Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, Saket Saurabh","doi":"10.1007/s00224-023-10136-w","DOIUrl":"https://doi.org/10.1007/s00224-023-10136-w","url":null,"abstract":"Abstract In the study of parameterized streaming complexity on graph problems, the main goal is to design streaming algorithms for parameterized problems such that $$mathcal {O}(f(k) log ^{mathcal {O}(1)} n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mo>log</mml:mo> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> space is enough, where f is an arbitrary computable function depending only on the parameter k . However, in the past few years very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size k of the solution we are seeking. Chitnis et al. (SODA’16) have shown that many important parameterized problems that form the backbone of traditional parameterized complexity are known to require $$Omega (n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> bits of storage for any streaming algorithm; e.g. Feedback Vertex Set , Even Cycle Transversal , Odd Cycle Transversal , Triangle Deletion or the more general $$mathcal{F}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>F</mml:mi> </mml:math> - Subgraph Deletion when parameterized by solution size k . Our contribution lies in overcoming the obstacles to efficient parameterized streaming algorithms in graph deletion problems by utilizing the power of parameterization. We focus on the vertex cover size K as the parameter for the parameterized graph deletion problems we consider. In this work, we consider the four most well-studied streaming models: the Ea , Dea , Va (vertex arrival) and Al (adjacency list) models. Surprisingly, the consideration of vertex cover size K in the different models leads to a classification of positive and negative results for problems like $$mathcal{F}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>F</mml:mi> </mml:math> - Subgraph Deletion and $$mathcal{F}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>F</mml:mi> </mml:math> - Minor Deletion .","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136308259","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 : 2023-08-28DOI: 10.1007/s00224-023-10140-0
Vadim E. Levit, David Tankus
{"title":"Well-Covered Graphs With Constraints On $$Delta $$ And $$delta $$","authors":"Vadim E. Levit, David Tankus","doi":"10.1007/s00224-023-10140-0","DOIUrl":"https://doi.org/10.1007/s00224-023-10140-0","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46652296","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 : 2023-08-28DOI: 10.1007/s00224-023-10143-x
Dariusz R. Kowalski, Miguel A. Mosteiro, Kevin Zaki
{"title":"Correction to: Dynamic Multiple-Message Broadcast: Bounding Throughput in the Affectance Model","authors":"Dariusz R. Kowalski, Miguel A. Mosteiro, Kevin Zaki","doi":"10.1007/s00224-023-10143-x","DOIUrl":"https://doi.org/10.1007/s00224-023-10143-x","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42833829","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 : 2023-08-25DOI: 10.1007/s00224-023-10130-2
Henning Fernau, Kshitij Gajjar
Abstract A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every n -vertex sum graph can be represented by a sorted list of n positive integers where edge queries can be answered in $$mathscr {O}(log n)$$ O(logn) time. Therefore, upper-bounding the numbers used as vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every n -vertex, m -edge, d -degenerate graph can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as $$mathscr {O}(n^2d)$$ O(n2d) . This enables us to store the graph using $$mathscr {O}(mlog n)$$ O(mlogn) bits of memory. For sparse graphs (graphs with $$mathscr {O}(n)$$ O(n) edges), this matches the trivial lower bound of $$Omega (nlog n)$$ Ω(nlogn) . As planar graphs and forests have constant degeneracy, our result implies an upper bound of $$mathscr {O}(n^2)$$ O(n2) on their label numbers. The previously best known upper bound on the numbers needed for labelling general graphs with the minimum number of isolated vertices was $$mathscr {O}(4^n)$$ O(4n) , due to Kratochvíl, Miller & Nguyen (2001). Furthermore,
当且仅当两个顶点的标记之和为图中另一个顶点的标记时，两个顶点之间存在一条边，并可以用不同的正整数来标记，则图称为和图。大多数关于和图的论文考虑的是组合问题，比如需要将孤立顶点的最小数量添加到给定图中以使其成为和图。本文从计算复杂性的角度出发，对和图进行了研究。注意，每个n顶点和图都可以用n个正整数的排序列表表示，其中边查询可以在$$mathscr {O}(log n)$$ O (log n)时间内得到回答。因此，作为顶点标签的数字的上限也限制了在数据库中存储图的空间复杂度。我们证明了每个n顶点，m边，d退化图都可以通过向其添加最多m个孤立顶点来构成求和图，这样用作顶点标签的最大数字增长为$$mathscr {O}(n^2d)$$ O (n 2d)。这使我们能够使用$$mathscr {O}(mlog n)$$ O (m log n)位内存来存储图形。对于稀疏图(具有$$mathscr {O}(n)$$ O (n)条边的图)，这与$$Omega (nlog n)$$ Ω (n log n)的平凡下界相匹配。由于平面图和森林具有恒定的简并性，我们的结果表明它们的标号的上界为$$mathscr {O}(n^2)$$ O (n 2)。先前已知的标记具有最小孤立顶点数的一般图所需的数的上界是$$mathscr {O}(4^n)$$ O (4 n)，由于Kratochvíl, Miller ＆Nguyen(2001)。此外，他们的证明是存在的，而我们的标记可以在多项式时间内构造。
{"title":"The Space Complexity of Sum Labelling","authors":"Henning Fernau, Kshitij Gajjar","doi":"10.1007/s00224-023-10130-2","DOIUrl":"https://doi.org/10.1007/s00224-023-10130-2","url":null,"abstract":"Abstract A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every n -vertex sum graph can be represented by a sorted list of n positive integers where edge queries can be answered in $$mathscr {O}(log n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> time. Therefore, upper-bounding the numbers used as vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every n -vertex, m -edge, d -degenerate graph can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as $$mathscr {O}(n^2d)$$ <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>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . This enables us to store the graph using $$mathscr {O}(mlog n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> bits of memory. For sparse graphs (graphs with $$mathscr {O}(n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> edges), this matches the trivial lower bound of $$Omega (nlog n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . As planar graphs and forests have constant degeneracy, our result implies an upper bound of $$mathscr {O}(n^2)$$ <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>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> on their label numbers. The previously best known upper bound on the numbers needed for labelling general graphs with the minimum number of isolated vertices was $$mathscr {O}(4^n)$$ <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:mn>4</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , due to Kratochvíl, Miller & Nguyen (2001). Furthermore, ","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135236029","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 : 2023-08-23DOI: 10.1007/s00224-023-10137-9
David Furcy, Scott M. Summers, Logan Withers
{"title":"Improved Lower and Upper Bounds on the Tile Complexity of Uniquely Self-Assembling a Thin Rectangle Non-Cooperatively in 3D","authors":"David Furcy, Scott M. Summers, Logan Withers","doi":"10.1007/s00224-023-10137-9","DOIUrl":"https://doi.org/10.1007/s00224-023-10137-9","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135520267","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 : 2023-08-15DOI: 10.1007/s00224-023-10141-z
Vladan Gloncak, Jarl Emil Erla Munkstrup, Jakob Grue Simonsen
{"title":"Implicit Representation of Relations","authors":"Vladan Gloncak, Jarl Emil Erla Munkstrup, Jakob Grue Simonsen","doi":"10.1007/s00224-023-10141-z","DOIUrl":"https://doi.org/10.1007/s00224-023-10141-z","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47723134","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 : 2023-08-12DOI: 10.1007/s00224-023-10135-x
Jaroslav Garvardt, Christian Komusiewicz, Frank Sommer
Abstract The s - Club problem asks whether a given undirected graph G contains a vertex set S of size at least k such that G [ S ], the subgraph of G induced by S , has diameter at most s . We consider variants of s - Club where one additionally demands that each vertex of G [ S ] is contained in at least $$ell $$ ℓ triangles in G [ S ], that each edge of G [ S ] is contained in at least $$ell $$ ℓ triangles in G [ S ], or that S contains a given set W of seed vertices. We show that in general these variants are W[1]-hard when parameterized by the solution size k , making them significantly harder than the unconstrained s - Club problem. On the positive side, we obtain some FPT algorithms for the case when $$ell =1$$ ℓ=1 and for the case when G [ W ], the graph induced by the set of seed vertices, is a clique.
{"title":"The Parameterized Complexity of s-Club with Triangle and Seed Constraints","authors":"Jaroslav Garvardt, Christian Komusiewicz, Frank Sommer","doi":"10.1007/s00224-023-10135-x","DOIUrl":"https://doi.org/10.1007/s00224-023-10135-x","url":null,"abstract":"Abstract The s - Club problem asks whether a given undirected graph G contains a vertex set S of size at least k such that G [ S ], the subgraph of G induced by S , has diameter at most s . We consider variants of s - Club where one additionally demands that each vertex of G [ S ] is contained in at least $$ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> triangles in G [ S ], that each edge of G [ S ] is contained in at least $$ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> triangles in G [ S ], or that S contains a given set W of seed vertices. We show that in general these variants are W[1]-hard when parameterized by the solution size k , making them significantly harder than the unconstrained s - Club problem. On the positive side, we obtain some FPT algorithms for the case when $$ell =1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and for the case when G [ W ], the graph induced by the set of seed vertices, is a clique.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134977843","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 : 2023-08-11DOI: 10.1007/s00224-023-10138-8
Andreas Darmann
{"title":"Stability and Welfare in (Dichotomous) Hedonic Diversity Games","authors":"Andreas Darmann","doi":"10.1007/s00224-023-10138-8","DOIUrl":"https://doi.org/10.1007/s00224-023-10138-8","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47414897","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 : 2023-07-26DOI: 10.1007/s00224-023-10139-7
Jarkko Kari
Abstract A d -dimensional configuration $$c:mathbb {Z}^dlongrightarrow A$$ c:Zd⟶A is a coloring of the d -dimensional infinite grid by elements of a finite alphabet $$Asubseteq mathbb {Z}$$ A⊆Z . The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d -variate formal power series, the annihilator is conveniently expressed as a d -variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c . A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d -dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a $$(d-1)$$ (d-1) -dimensional linear subspace $$Ssubseteq mathbb {R}^d$$ S⊆Rd is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S . As a subshift is known to be finite if all $$(d-1)$$ (d-1) -dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of $$mathbb {Z}^d$$ Zd by translations of a single tile.
d维构形$$c:mathbb {Z}^dlongrightarrow A$$ c: Z d ? A是用有限字母的元素对d维无限网格的着色$$Asubseteq mathbb {Z}$$ A≠Z。构型c有湮灭子如果c的有限多个平移的非平凡线性组合是零构型。将c写成d变量形式幂级数，湮灭子可以方便地表示为d变量洛朗多项式f，它与c的形式积是零幂级数。更一般地说，如果形式积是一个强周期构型，我们称多项式f为c的周期器。一组构型的公共湮灭子(周期子)称为该集合的湮灭子(分别为周期子)。特别地，我们考虑了d维子位移的湮灭子和周期子，即通过不允许某些局部模式定义的组态集。我们证明了$$(d-1)$$ (d - 1)维线性子空间$$Ssubseteq mathbb {R}^d$$ S∈R d对于子位移是可扩展的，如果子位移有一个周期器，其支撑只包含S的一个元素。如果所有$$(d-1)$$ (d - 1)维子空间都是可扩张的，则子位移是有限的，我们在周期器上得到了保证子位移有限的一个简单必要条件，或者等价地，保证构型的强周期性。我们提供了通过翻译单个瓷砖来平铺$$mathbb {Z}^d$$ Z d的示例。
{"title":"Expansivity and Periodicity in Algebraic Subshifts","authors":"Jarkko Kari","doi":"10.1007/s00224-023-10139-7","DOIUrl":"https://doi.org/10.1007/s00224-023-10139-7","url":null,"abstract":"Abstract A d -dimensional configuration $$c:mathbb {Z}^dlongrightarrow A$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>⟶</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> is a coloring of the d -dimensional infinite grid by elements of a finite alphabet $$Asubseteq mathbb {Z}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> . The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d -variate formal power series, the annihilator is conveniently expressed as a d -variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c . A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d -dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a $$(d-1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional linear subspace $$Ssubseteq mathbb {R}^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S . As a subshift is known to be finite if all $$(d-1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of $$mathbb {Z}^d$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> by translations of a single tile.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135800782","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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