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":"72 1","pages":"0"},"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":" ","pages":""},"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":"47 1","pages":"0"},"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}
Pub Date : 2023-07-23DOI: 10.1007/s00224-023-10124-0
Jozef Jirásek, Ian McQuillan
An automaton is k-visit-bounded if during any computation its work tape head visits each tape cell at most k times. In this paper we consider stack automata which are k-visit-bounded for some integer k. This restriction resets the visits when popping (unlike similarly defined Turing machine restrictions) which we show allows the model to accept a proper superset of context-free languages and also a proper superset of languages of visit-bounded Turing machines. We study two variants of visit-bounded stack automata: one where only instructions that move the stack head downwards increase the number of visits of the destination cell, and another where any transition increases the number of visits. We prove that the two types of automata recognize the same languages. We then show that all languages recognized by visit-bounded stack automata are effectively semilinear, and hence are letter-equivalent to regular languages, which can be used to show other properties.
{"title":"Visit-Bounded Stack Automata","authors":"Jozef Jirásek, Ian McQuillan","doi":"10.1007/s00224-023-10124-0","DOIUrl":"https://doi.org/10.1007/s00224-023-10124-0","url":null,"abstract":"<p>An automaton is <i>k-visit-bounded</i> if during any computation its work tape head visits each tape cell at most <i>k</i> times. In this paper we consider stack automata which are <i>k</i>-visit-bounded for some integer <i>k</i>. This restriction resets the visits when popping (unlike similarly defined Turing machine restrictions) which we show allows the model to accept a proper superset of context-free languages and also a proper superset of languages of visit-bounded Turing machines. We study two variants of visit-bounded stack automata: one where only instructions that move the stack head downwards increase the number of visits of the destination cell, and another where any transition increases the number of visits. We prove that the two types of automata recognize the same languages. We then show that all languages recognized by visit-bounded stack automata are effectively semilinear, and hence are letter-equivalent to regular languages, which can be used to show other properties.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"182 ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514140","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 : 2023-07-12DOI: 10.1007/s00224-023-10134-y
K. Subramani, P. Wojciechowski
{"title":"Unit Read-once Refutations for Systems of Difference Constraints","authors":"K. Subramani, P. Wojciechowski","doi":"10.1007/s00224-023-10134-y","DOIUrl":"https://doi.org/10.1007/s00224-023-10134-y","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"877 - 899"},"PeriodicalIF":0.5,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41737660","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 : 2023-06-01DOI: 10.1007/s00224-023-10123-1
E. Mayordomo, M. Ogihara, A. Rudra
{"title":"Foreword: a Commemorative Issue for Alan L. Selman","authors":"E. Mayordomo, M. Ogihara, A. Rudra","doi":"10.1007/s00224-023-10123-1","DOIUrl":"https://doi.org/10.1007/s00224-023-10123-1","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"415 - 416"},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42059463","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 : 2023-05-27DOI: 10.1007/s00224-023-10128-w
Rustem Takhanov
{"title":"Computing a Partition Function of a Generalized Pattern-Based Energy over a Semiring","authors":"Rustem Takhanov","doi":"10.1007/s00224-023-10128-w","DOIUrl":"https://doi.org/10.1007/s00224-023-10128-w","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"67 1","pages":"760 - 784"},"PeriodicalIF":0.5,"publicationDate":"2023-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41667781","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 : 2023-05-09DOI: 10.4230/LIPIcs.STACS.2021.51
Markus Lohrey
It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time when all group elements are represented by so-called power words, i.e., words of the form $$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ p 1 z 1 p 2 z 2 ⋯ p k z k . Here the $$p_i$$ p i are explicit words over the generating set of the group and all $$z_i$$ z i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group $$textsf{GL}(2,mathbb {Z})$$ GL ( 2 , Z ) can be decided in polynomial time when elements of $$textsf{GL}(2,mathbb {Z})$$ GL ( 2 , Z ) are represented by matrices with binary encoded integers. For the same input representation, it also shown that one can compute in polynomial time the index of a given finitely generated subgroup of $$textsf{GL}(2,mathbb {Z})$$ GL ( 2 , Z ) .
结果表明，当所有群元素都由所谓的幂词表示时，一个几乎自由的群的子群隶属问题可以在多项式时间内确定，即$$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ p 1 z 1 p 2 z 2⋯p k z k。这里的$$p_i$$ pi是组的生成集上的显式单词，所有的$$z_i$$ zi都是二进制编码的整数。作为推论，当$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的元素用二进制编码的整数矩阵表示时，矩阵群$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的子群隶属性问题可以在多项式时间内确定。对于相同的输入表示，它还表明可以在多项式时间内计算给定的有限生成的子群$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的索引。
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Pub Date : 2023-05-09DOI: 10.1007/s00224-023-10122-2
Markus Lohrey
Abstract It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time when all group elements are represented by so-called power words, i.e., words of the form $$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ p1z1p2z2⋯pkzk . Here the $$p_i$$ pi are explicit words over the generating set of the group and all $$z_i$$ zi are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group $$textsf{GL}(2,mathbb {Z})$$ GL(2,Z) can be decided in polynomial time when elements of $$textsf{GL}(2,mathbb {Z})$$ GL(2,Z) are represented by matrices with binary encoded integers. For the same input representation, it also shown that one can compute in polynomial time the index of a given finitely generated subgroup of $$textsf{GL}(2,mathbb {Z})$$ GL(2,Z) .
摘要证明了一个几乎自由群的子群隶属问题可以在多项式时间内决定，当所有群元素都由所谓的幂词表示时，即$$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ p 1 z 1 p 2 z 2⋯p k z k。这里的$$p_i$$ pi是组的生成集上的显式单词，所有的$$z_i$$ zi都是二进制编码的整数。作为推论，当$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的元素用二进制编码的整数矩阵表示时，矩阵群$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的子群隶属性问题可以在多项式时间内确定。对于相同的输入表示，它还表明可以在多项式时间内计算给定的有限生成的子群$$textsf{GL}(2,mathbb {Z})$$ GL (2, Z)的索引。
{"title":"Subgroup Membership in GL(2,Z)","authors":"Markus Lohrey","doi":"10.1007/s00224-023-10122-2","DOIUrl":"https://doi.org/10.1007/s00224-023-10122-2","url":null,"abstract":"Abstract It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time when all group elements are represented by so-called power words, i.e., words of the form $$p_1^{z_1} p_2^{z_2} cdots p_k^{z_k}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mn>1</mml:mn> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:msubsup> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:msubsup> <mml:mo>⋯</mml:mo> <mml:msubsup> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msubsup> </mml:mrow> </mml:math> . Here the $$p_i$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> are explicit words over the generating set of the group and all $$z_i$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group $$textsf{GL}(2,mathbb {Z})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be decided in polynomial time when elements of $$textsf{GL}(2,mathbb {Z})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are represented by matrices with binary encoded integers. For the same input representation, it also shown that one can compute in polynomial time the index of a given finitely generated subgroup of $$textsf{GL}(2,mathbb {Z})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135806947","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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