Pub Date : 2023-10-19DOI: 10.1007/s00224-023-10147-7
David E. Brown, David Skidmore
{"title":"Representing the Integer Factorization Problem Using Ordered Binary Decision Diagrams","authors":"David E. Brown, David Skidmore","doi":"10.1007/s00224-023-10147-7","DOIUrl":"https://doi.org/10.1007/s00224-023-10147-7","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135667772","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 : 2023-10-06DOI: 10.1007/s00224-023-10145-9
Abdolhamid Ghodselahi, Fabian Kuhn
Abstract We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. (SODA 258–267 2007)). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of k mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics . A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the k -server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.
{"title":"Toward Online Mobile Facility Location on General Metrics","authors":"Abdolhamid Ghodselahi, Fabian Kuhn","doi":"10.1007/s00224-023-10145-9","DOIUrl":"https://doi.org/10.1007/s00224-023-10145-9","url":null,"abstract":"Abstract We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. (SODA 258–267 2007)). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of k mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics . A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the k -server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"3 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":"135350992","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 : 2023-10-03DOI: 10.1007/s00224-023-10144-w
Andreas Maletti, Andreea-Teodora Nász
Abstract The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Giménez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem remains open for regular weighted tree languages. It is demonstrated that the main notion used in the unweighted setting, the tree automaton with equality and inequality constraints , can straightforwardly be generalized to the weighted setting and can represent the image of any regular weighted tree language under any nondeleting and nonerasing tree homomorphism. Several closure properties as well as decision problems are also investigated for the weighted tree languages generated by weighted tree automata with constraints.
{"title":"Weighted Tree Automata with Constraints","authors":"Andreas Maletti, Andreea-Teodora Nász","doi":"10.1007/s00224-023-10144-w","DOIUrl":"https://doi.org/10.1007/s00224-023-10144-w","url":null,"abstract":"Abstract The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Giménez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem remains open for regular weighted tree languages. It is demonstrated that the main notion used in the unweighted setting, the tree automaton with equality and inequality constraints , can straightforwardly be generalized to the weighted setting and can represent the image of any regular weighted tree language under any nondeleting and nonerasing tree homomorphism. Several closure properties as well as decision problems are also investigated for the weighted tree languages generated by weighted tree automata with constraints.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135697010","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 : 2023-09-23DOI: 10.1007/s00224-023-10132-0
Lars Jaffke, Paloma T. Lima, Daniel Lokshtanov
Abstract We provide a polynomial-time algorithm for b -Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva (Algorithmica 80 (1), 104–115, 2018) and Bonomo et al. (Graphs and Combinatorics 25 (2), 153–167, 2009). This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is $$textsf{FPT}$$ FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for $$b$$ b - Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.
摘要给出了常团宽图上b -上色的多项式时间算法。这统一并扩展了几乎所有已知的图类多项式时间结果，并回答了Campos和Silva (Algorithmica 80(1)， 104-115, 2018)和Bonomo等人(Graphs and Combinatorics 25(2)， 153-167, 2009)提出的开放问题。这是关于这个问题的结构参数化的第一个结果。我们证明了在一般图上用顶点覆盖数参数化的问题是$$textsf{FPT}$$ FPT，在弦图上用颜色数参数化的问题是 FPT。此外，我们观察到我们的有界团宽度图的算法可以在相同的运行时间范围内适用于解决Fall Coloring问题。在指数时间假设下，基于团宽度的$$b$$ b -着色和Fall着色算法的运行时间较紧。
{"title":"b-Coloring Parameterized by Clique-Width","authors":"Lars Jaffke, Paloma T. Lima, Daniel Lokshtanov","doi":"10.1007/s00224-023-10132-0","DOIUrl":"https://doi.org/10.1007/s00224-023-10132-0","url":null,"abstract":"Abstract We provide a polynomial-time algorithm for b -Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva (Algorithmica 80 (1), 104–115, 2018) and Bonomo et al. (Graphs and Combinatorics 25 (2), 153–167, 2009). This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is $$textsf{FPT}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>FPT</mml:mi> </mml:math> when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for $$b$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>b</mml:mi> </mml:math> - Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135959829","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}
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":"9 1","pages":"0"},"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":"1 1","pages":""},"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":"67 1","pages":"1131 - 1131"},"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 $$mathscr {O}(log n)$$ <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,
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":"24 1","pages":"0"},"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":"61 1","pages":"0"},"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":" ","pages":""},"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}
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