Pub Date : 2024-10-29DOI: 10.1016/j.tcs.2024.114939
Thierry Boy de la Tour
Monographs are graph-like structures with directed edges of unlimited length that are freely adjacent to each other. The standard nodes are represented as edges of length zero. They can be drawn in a way consistent with standard graphs and many others, like E-graphs or ∞-graphs. The category of monographs share many properties with the categories of graph structures (algebras of monadic many-sorted signatures, equivalent to presheaf toposes), except that there is no terminal monograph. It is universal in the sense that its slice categories (or categories of typed monographs) are equivalent to the categories of graph structures. Type monographs thus emerge as a natural way of specifying graph structures. A detailed analysis of single and double pushout transformations of monographs is provided, and a notion of attributed typed monographs generalizing typed attributed E-graphs is analyzed w.r.t. attribute-preserving transformations.
专著是一种类似图的结构，其有向边的长度不受限制，可以自由地相互毗邻。标准节点用长度为零的边表示。它们的绘制方式与标准图和许多其他图（如 E 图或∞图）一致。单论范畴与图结构范畴（单论多排序签名代数，等同于预设拓扑）有许多相同的性质，只是没有终端单论。从它的切片范畴（或类型专著范畴）等同于图结构范畴的意义上讲，它是通用的。因此，类型专著是指定图结构的一种自然方式。本文对专著的单推出和双推出变换进行了详细分析，并结合属性保留变换分析了归属类型专著的概念，该概念概括了类型化归属电子图。
{"title":"Algebraic properties and transformations of monographs","authors":"Thierry Boy de la Tour","doi":"10.1016/j.tcs.2024.114939","DOIUrl":"10.1016/j.tcs.2024.114939","url":null,"abstract":"<div><div>Monographs are graph-like structures with directed edges of unlimited length that are freely adjacent to each other. The standard nodes are represented as edges of length zero. They can be drawn in a way consistent with standard graphs and many others, like E-graphs or ∞-graphs. The category of monographs share many properties with the categories of graph structures (algebras of monadic many-sorted signatures, equivalent to presheaf toposes), except that there is no terminal monograph. It is universal in the sense that its slice categories (or categories of typed monographs) are equivalent to the categories of graph structures. Type monographs thus emerge as a natural way of specifying graph structures. A detailed analysis of single and double pushout transformations of monographs is provided, and a notion of attributed typed monographs generalizing typed attributed E-graphs is analyzed w.r.t. attribute-preserving transformations.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114939"},"PeriodicalIF":0.9,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553817","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}
We analyze the Disjoint Path Allocation problem (DPA) in the priority framework. Motivated by the problem of traffic regulation in communication networks, DPA consists of allocating edge-disjoint paths in a graph. Like an online algorithm, a priority algorithm receives its input sequentially and must output irrevocable decisions for individual input items before having seen the entire input. However, in contrast to the online setting, a priority algorithm may choose an order on the set of all possible input items and the actual input is then presented according to this order. A priority algorithm is thus a natural model for the intuitively well-understood concept of a greedy algorithm.
Mainly motivated by their application for proving lower bounds, we also consider priority algorithms with advice, thus measuring the necessary amount of information about the yet unknown parts of the input.
Besides considering the classical variant of the DPA problem on paths and the related problem of Length-Weighted DPA, we mainly focus on DPA on trees. We show asymptotically matching upper and lower bounds on the advice necessary for optimality in LWDPA and generalize the known optimality result for DPA on paths to trees with maximal degree at most 3. On trees with higher maximal degree, we prove matching upper and lower bounds on the approximation ratio in the advice-free priority setting as well as upper and lower bounds on the advice necessary to achieve optimality.
{"title":"Priority algorithms with advice for disjoint path allocation problems","authors":"Hans-Joachim Böckenhauer , Fabian Frei , Silvan Horvath","doi":"10.1016/j.tcs.2024.114942","DOIUrl":"10.1016/j.tcs.2024.114942","url":null,"abstract":"<div><div>We analyze the Disjoint Path Allocation problem (DPA) in the priority framework. Motivated by the problem of traffic regulation in communication networks, DPA consists of allocating edge-disjoint paths in a graph. Like an online algorithm, a priority algorithm receives its input sequentially and must output irrevocable decisions for individual input items before having seen the entire input. However, in contrast to the online setting, a priority algorithm may choose an order on the set of all possible input items and the actual input is then presented according to this order. A priority algorithm is thus a natural model for the intuitively well-understood concept of a greedy algorithm.</div><div>Mainly motivated by their application for proving lower bounds, we also consider priority algorithms with advice, thus measuring the necessary amount of information about the yet unknown parts of the input.</div><div>Besides considering the classical variant of the DPA problem on paths and the related problem of Length-Weighted DPA, we mainly focus on DPA on trees. We show asymptotically matching upper and lower bounds on the advice necessary for optimality in LWDPA and generalize the known optimality result for DPA on paths to trees with maximal degree at most 3. On trees with higher maximal degree, we prove matching upper and lower bounds on the approximation ratio in the advice-free priority setting as well as upper and lower bounds on the advice necessary to achieve optimality.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114942"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-28DOI: 10.1016/j.tcs.2024.114937
Jean-Éric Pin
Let be an integer. For each positive integer d, let be the Euclidean division by d in base b, that is, the function which associates to a word u in , representing an integer n in base b, the unique word of the same length as u representing the quotient of the division of n by d. We describe the pure sequential transducer realizing this function and analyze the algebraic structure of its syntactic monoid. We compute its size, describe its Green's relations and its minimum ideal. As a consequence, we show that it is a group if and only if d and b are coprime numbers, it is a p-group if and only if d is a power of p and b is congruent to 1 modulo p and it is an aperiodic monoid if and only if d divides some power of b. The uniform continuity of for the pro-group metric was studied by Reutenauer and Schützenberger in 1995. We launch a similar study for the uniform continuity of with respect to the pro-p metric, where p is a prime number.
设 b⩾2 为整数。对于每个正整数 d，设 Ed,b 是以 b 为底的欧几里得除法，即把代表以 b 为底的整数 n 的 {0,...,b-1}⁎ 中的单词 u 与代表 n 除以 d 的商的与 u 长度相同的唯一单词关联起来的函数。我们计算了它的大小，描述了它的格林关系和最小理想数。因此，我们证明了当且仅当 d 和 b 是同素数时，它是一个群；当且仅当 d 是 p 的幂和 b 同调于 1 modulo p 时，它是一个 p 群；当且仅当 d 除以 b 的某个幂时，它是一个非周期性单元。我们对 Ed,b 关于亲群公设（其中 p 是素数）的均匀连续性进行了类似的研究。
{"title":"Euclidean division by d in base b","authors":"Jean-Éric Pin","doi":"10.1016/j.tcs.2024.114937","DOIUrl":"10.1016/j.tcs.2024.114937","url":null,"abstract":"<div><div>Let <span><math><mi>b</mi><mo>⩾</mo><mn>2</mn></math></span> be an integer. For each positive integer <em>d</em>, let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> be the Euclidean division by <em>d</em> in base <em>b</em>, that is, the function which associates to a word <em>u</em> in <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, representing an integer <em>n</em> in base <em>b</em>, the unique word of the same length as <em>u</em> representing the quotient of the division of <em>n</em> by <em>d</em>. We describe the pure sequential transducer realizing this function and analyze the algebraic structure of its syntactic monoid. We compute its size, describe its Green's relations and its minimum ideal. As a consequence, we show that it is a group if and only if <em>d</em> and <em>b</em> are coprime numbers, it is a <em>p</em>-group if and only if <em>d</em> is a power of <em>p</em> and <em>b</em> is congruent to 1 modulo <em>p</em> and it is an aperiodic monoid if and only if <em>d</em> divides some power of <em>b</em>. The uniform continuity of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> for the pro-group metric was studied by Reutenauer and Schützenberger in 1995. We launch a similar study for the uniform continuity of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> with respect to the pro-<em>p</em> metric, where <em>p</em> is a prime number.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114937"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553818","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 : 2024-10-28DOI: 10.1016/j.tcs.2024.114931
Yin Liu, Austen Z. Fan, Jin-Yi Cai
denotes a class of counting problems specified by a constraint function f. We prove complexity dichotomy theorems for in two settings: (1) f is any symmetric arity-3 real-valued function on input of domain size 3. (2) f is any symmetric arity-3 -valued function on input of domain size 4.
Holant⁎(f) 表示一类由约束函数 f 指定的计数问题。我们将在两种情况下证明 Holant⁎(f) 的复杂性二分定理：(1) f 是域大小为 3 的输入上的任意对称算术值-3 实值函数；(2) f 是域大小为 4 的输入上的任意对称算术值-3 {0,1} 值函数。
{"title":"Restricted holant dichotomy on domain sizes 3 and 4","authors":"Yin Liu, Austen Z. Fan, Jin-Yi Cai","doi":"10.1016/j.tcs.2024.114931","DOIUrl":"10.1016/j.tcs.2024.114931","url":null,"abstract":"<div><div><span><math><msup><mrow><mi>Holant</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>f</mi><mo>)</mo></math></span> denotes a class of counting problems specified by a constraint function <em>f</em>. We prove complexity dichotomy theorems for <span><math><msup><mrow><mi>Holant</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>f</mi><mo>)</mo></math></span> in two settings: (1) <em>f</em> is any symmetric arity-3 real-valued function on input of domain size 3. (2) <em>f</em> is any symmetric arity-3 <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span>-valued function on input of domain size 4.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1023 ","pages":"Article 114931"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553640","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 : 2024-10-28DOI: 10.1016/j.tcs.2024.114938
Barkha Singh, Sreedevi Indu, Sudipta Majumdar
This article analyzes and compares various Quantum machine learning algorithms on big data. The main contribution of this article is to provide a new machine-learning approach using Quantum computing for big data analysis with features of robust, novel, and effective Quantum computing. This work proposes a global Quantum feature extraction technique for large-scale image classification based on Schmidt decomposition for the first time. Additionally, a new version of the Quantum learning algorithm is presented, which uses the features of Hamming distance to classify images. With the help of algorithm analysis and experimental findings from the benchmark database Caltech 101, a successful method for large-scale image classification is developed and put forth in the context of big data. The proposed model yields an average accuracy of 98% with the proposed enhanced Quantum classifier, QeSVM classifier, swarm particle optimizer with Twin wave SVM, QPSO-TWSVM, and other Q-CNN models on different Big Data sets.
{"title":"Comparison of machine learning algorithms for classification of Big Data sets","authors":"Barkha Singh, Sreedevi Indu, Sudipta Majumdar","doi":"10.1016/j.tcs.2024.114938","DOIUrl":"10.1016/j.tcs.2024.114938","url":null,"abstract":"<div><div>This article analyzes and compares various Quantum machine learning algorithms on big data. The main contribution of this article is to provide a new machine-learning approach using Quantum computing for big data analysis with features of robust, novel, and effective Quantum computing. This work proposes a global Quantum feature extraction technique for large-scale image classification based on Schmidt decomposition for the first time. Additionally, a new version of the Quantum learning algorithm is presented, which uses the features of Hamming distance to classify images. With the help of algorithm analysis and experimental findings from the benchmark database Caltech 101, a successful method for large-scale image classification is developed and put forth in the context of big data. The proposed model yields an average accuracy of 98% with the proposed enhanced Quantum classifier, QeSVM classifier, swarm particle optimizer with Twin wave SVM, QPSO-TWSVM, and other Q-CNN models on different Big Data sets.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114938"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586215","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 : 2024-10-28DOI: 10.1016/j.tcs.2024.114923
Anna Nenca , Barbara Wolnik , Bernard De Baets
The parity problem is one of the best-known classification problems studied to examine the computational abilities of cellular automata. In this inverse problem, one is looking for a cellular automaton that can classify each initial configuration into one of two classes according to its parity. In the case of deterministic one-dimensional cellular automata, there exists a local rule that effectively solves the parity problem, but it is unknown whether it is the simplest possible rule. Specifically, it is known that a cellular automaton with a nine-cell neighborhood can solve the parity problem, whereas no cellular automaton with a five-cell neighborhood is capable of doing so. These findings have remained unimproved for the past 10 years. In this paper, we present novel tools that allow to narrow down the existing gap. With the help of these tools, we are able to demonstrate that there is no cellular automaton with a six-cell neighborhood capable of solving the parity problem.
{"title":"No six-cell neighborhood cellular automaton solves the parity problem","authors":"Anna Nenca , Barbara Wolnik , Bernard De Baets","doi":"10.1016/j.tcs.2024.114923","DOIUrl":"10.1016/j.tcs.2024.114923","url":null,"abstract":"<div><div>The parity problem is one of the best-known classification problems studied to examine the computational abilities of cellular automata. In this inverse problem, one is looking for a cellular automaton that can classify each initial configuration into one of two classes according to its parity. In the case of deterministic one-dimensional cellular automata, there exists a local rule that effectively solves the parity problem, but it is unknown whether it is the simplest possible rule. Specifically, it is known that a cellular automaton with a nine-cell neighborhood can solve the parity problem, whereas no cellular automaton with a five-cell neighborhood is capable of doing so. These findings have remained unimproved for the past 10 years. In this paper, we present novel tools that allow to narrow down the existing gap. With the help of these tools, we are able to demonstrate that there is no cellular automaton with a six-cell neighborhood capable of solving the parity problem.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114923"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554208","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}
Motivated by the recent trend of increasing number of e-cars and hybrid cars, we investigate the problem of building a red-black spanner for a mixed-charging vehicular network. In such a network, we have two kinds of gas/charging stations: electric (black) and the traditional gas (red) stations. Our requirement is that one cannot connect two gas stations directly in the spanner (i.e., no red-red edge), and the goal is to build a linear-size spanner with a bounded stretch factor under this requirement. (In 2-d, it can be shown that a spanner with an optimal stretch factor could have a quadratic size and if one is constrained to build the spanner purely from a given road network then it is impossible to obtain a bounded stretch factor.) Our main results are summarized as follows.
1.
In 1-d, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves the optimal stretch factor.
2.
In 2-d and under the metric, we build a linear-size red-black spanner satisfying the ‘no red-red edge’ requirement which achieves a stretch factor of 1.998.
3.
In 2-d and under the metric, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves a stretch factor of 3.613.
{"title":"Constructing red-black spanners for mixed-charging vehicular networks","authors":"Sergey Bereg , Yuya Higashikawa , Naoki Katoh , Junichi Teruyama , Yuki Tokuni , Binhai Zhu","doi":"10.1016/j.tcs.2024.114932","DOIUrl":"10.1016/j.tcs.2024.114932","url":null,"abstract":"<div><div>Motivated by the recent trend of increasing number of e-cars and hybrid cars, we investigate the problem of building a red-black spanner for a mixed-charging vehicular network. In such a network, we have two kinds of gas/charging stations: electric (black) and the traditional gas (red) stations. Our requirement is that one cannot connect two gas stations directly in the spanner (i.e., no red-red edge), and the goal is to build a linear-size spanner with a bounded stretch factor under this requirement. (In 2-d, it can be shown that a spanner with an optimal stretch factor could have a quadratic size and if one is constrained to build the spanner purely from a given road network then it is impossible to obtain a bounded stretch factor.) Our main results are summarized as follows.<ul><li><span>1.</span><span><div>In 1-d, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves the optimal stretch factor.</div></span></li><li><span>2.</span><span><div>In 2-d and under the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> metric, we build a linear-size red-black spanner satisfying the ‘no red-red edge’ requirement which achieves a stretch factor of 1.998.</div></span></li><li><span>3.</span><span><div>In 2-d and under the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> metric, a linear-size red-black spanner is built to satisfy the ‘no red-red edge’ requirement which achieves a stretch factor of 3.613.</div></span></li></ul></div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1023 ","pages":"Article 114932"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553639","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 : 2024-10-28DOI: 10.1016/j.tcs.2024.114928
Jakob Baumann, Matthias Pfretzschner, Ignaz Rutter
Motivated by the planarization of 2-layered straight-line drawings, we consider the problem of modifying a graph such that the resulting graph has pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks whether such a graph can be obtained using at most k vertex explosions, where a vertex explosion replaces a vertex v by degree-1 vertices, each incident to exactly one edge that was originally incident to v. For POVE, we give an FPT algorithm with running time and an kernel, thereby improving over the -kernel by Ahmed et al. [2] in a more general setting. Similarly, a vertex split replaces a vertex v by two distinct vertices and and distributes the edges originally incident to v arbitrarily to and . Analogously to POVE, we define the problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split operation instead of vertex explosions. Here we obtain a linear kernel and an algorithm with running time . This answers an open question by Ahmed et al. [2].
Finally, we consider the problem Π-VertexSplitting (Π-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class Π using at most k vertex splits. For graph classes Π that can be defined in monadic second-order graph logic (MSO2), we show that the problem Π-VS can be expressed as an MSO2 formula, resulting in an FPT algorithm for Π-VS parameterized by k if Π additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions.
受两层直线图平面化的启发，我们考虑了如何修改一个图，使得到的图的路径宽度至多为 1 的问题。路径宽度为 1 的顶点爆炸（POVE）问题问的是，是否能用至多 k 个顶点爆炸得到这样一个图，其中一个顶点爆炸用 deg(v) 度为 1 的顶点替换一个顶点 v，每个顶点都恰好与一条原来与 v 有关的边相连。对于 POVE，我们给出了运行时间为 O(4k⋅m)、内核为 O(k2)的 FPT 算法，从而改进了 Ahmed 等人[2]在更一般情况下的内核为 O(k6)的算法。同样，顶点拆分也是用两个不同的顶点 v1 和 v2 替换一个顶点 v，并将 v 原有的边任意分配给 v1 和 v2。与 POVE 类似，我们定义了问题变体 "路径宽度一顶点分割"（POVS），它使用分割操作代替顶点爆炸。在这里，我们得到了一个线性内核和运行时间为 O((6k+12)k⋅m)的算法。最后，我们考虑了 Π-VertexSplitting (Π-VS)问题，它是对 POVS 问题的概括，问的是给定的图是否能通过最多 k 个顶点分割变成特定图类 Π 的图。对于可以用一元二阶图逻辑（MSO2）定义的图类Π，我们证明了问题Π-VS 可以用 MSO2 公式来表达，如果Π 还具有有界树宽（treewidth），就能得到以 k 为参数的Π-VS 的 FPT 算法。对于使用顶点爆炸的问题变体，我们也得到了同样的结果。
{"title":"Parameterized complexity of vertex splitting to pathwidth at most 1","authors":"Jakob Baumann, Matthias Pfretzschner, Ignaz Rutter","doi":"10.1016/j.tcs.2024.114928","DOIUrl":"10.1016/j.tcs.2024.114928","url":null,"abstract":"<div><div>Motivated by the planarization of 2-layered straight-line drawings, we consider the problem of modifying a graph such that the resulting graph has pathwidth at most 1. The problem <span>Pathwidth-One Vertex Explosion</span> (<span>POVE</span>) asks whether such a graph can be obtained using at most <em>k</em> vertex explosions, where a <em>vertex explosion</em> replaces a vertex <em>v</em> by <span><math><mi>deg</mi><mo></mo><mo>(</mo><mi>v</mi><mo>)</mo></math></span> degree-1 vertices, each incident to exactly one edge that was originally incident to <em>v</em>. For <span>POVE</span>, we give an FPT algorithm with running time <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>⋅</mo><mi>m</mi><mo>)</mo></math></span> and an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> kernel, thereby improving over the <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span>-kernel by Ahmed et al. <span><span>[2]</span></span> in a more general setting. Similarly, a <em>vertex split</em> replaces a vertex <em>v</em> by two distinct vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and distributes the edges originally incident to <em>v</em> arbitrarily to <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Analogously to <span>POVE</span>, we define the problem variant <span>Pathwidth-One Vertex Splitting</span> (<span>POVS</span>) that uses the split operation instead of vertex explosions. Here we obtain a linear kernel and an algorithm with running time <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mn>6</mn><mi>k</mi><mo>+</mo><mn>12</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>⋅</mo><mi>m</mi><mo>)</mo></math></span>. This answers an open question by Ahmed et al. <span><span>[2]</span></span>.</div><div>Finally, we consider the problem Π<span>-VertexSplitting</span> (Π<span>-VS</span>), which generalizes the problem <span>POVS</span> and asks whether a given graph can be turned into a graph of a specific graph class Π using at most <em>k</em> vertex splits. For graph classes Π that can be defined in monadic second-order graph logic (MSO<sub>2</sub>), we show that the problem Π<span>-VS</span> can be expressed as an MSO<sub>2</sub> formula, resulting in an FPT algorithm for Π<span>-VS</span> parameterized by <em>k</em> if Π additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114928"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-28DOI: 10.1016/j.tcs.2024.114943
Junling Pei, Fang-Wei Fu
How to construct a non-interactive key exchange (NIKE) protocol based on coding theory is an opening problem. In this paper, we propose the first code-based NIKE protocol MODRED, whose security relies on the hardness of the Restricted Syndrome Decoding (R-SD) problem and its variant. A non-interactive error reconciliation mechanism is presented and applied to MODRED, which enables the two sides of communication to “approximately agree” on the shared key. This is of great significance because it may lead to a more generic framework construction. Furthermore, our proposal is competitive and practical. We provide an initial choice of parameters for MODRED, tailored to 120-bit post-quantum security level, yielding the public keys of less than 1.2 MBs.
{"title":"MODRED: A code-based non-interactive key exchange protocol","authors":"Junling Pei, Fang-Wei Fu","doi":"10.1016/j.tcs.2024.114943","DOIUrl":"10.1016/j.tcs.2024.114943","url":null,"abstract":"<div><div>How to construct a non-interactive key exchange (NIKE) protocol based on coding theory is an opening problem. In this paper, we propose the first code-based NIKE protocol MODRED, whose security relies on the hardness of the Restricted Syndrome Decoding (R-SD) problem and its variant. A non-interactive error reconciliation mechanism is presented and applied to MODRED, which enables the two sides of communication to “approximately agree” on the shared key. This is of great significance because it may lead to a more generic framework construction. Furthermore, our proposal is competitive and practical. We provide an initial choice of parameters for MODRED, tailored to 120-bit post-quantum security level, yielding the public keys of less than 1.2 MBs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114943"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554253","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 : 2024-10-28DOI: 10.1016/j.tcs.2024.114933
Xiaowei Li, Xiwen Lu
This paper considers a variant of the classic facility location game called the k-product facility location game with penalties. In this game, each facility can only produce at most one kind of product and each customer must be supplied with k different products or be rejected by paying the penalty cost. The authors propose a 2-approximate cross-monotonic cost-sharing scheme for the game when , and a ()-approximate cross-monotonic cost-sharing scheme when , respectively. Furthermore, the lower bound is obtained for the k-product facility location game with penalties.
本文研究的是经典设施选址博弈的一种变体，称为带惩罚的 k 种产品设施选址博弈。在这个博弈中，每个设施最多只能生产一种产品，而每个客户必须得到 k 种不同的产品，否则就要支付惩罚成本而被拒绝。作者分别提出了 k=2 时博弈的 2 近似交叉单调成本分摊方案和 k≥3 时的 (32k-32)- 近似交叉单调成本分摊方案。此外，还得到了 k 产品设施位置博弈的下界。
{"title":"An approximate cost recovery scheme for the k-product facility location game with penalties","authors":"Xiaowei Li, Xiwen Lu","doi":"10.1016/j.tcs.2024.114933","DOIUrl":"10.1016/j.tcs.2024.114933","url":null,"abstract":"<div><div>This paper considers a variant of the classic facility location game called the <em>k</em>-product facility location game with penalties. In this game, each facility can only produce at most one kind of product and each customer must be supplied with <em>k</em> different products or be rejected by paying the penalty cost. The authors propose a 2-approximate cross-monotonic cost-sharing scheme for the game when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, and a (<span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>k</mi><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>)-approximate cross-monotonic cost-sharing scheme when <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, respectively. Furthermore, the lower bound is obtained for the <em>k</em>-product facility location game with penalties.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114933"},"PeriodicalIF":0.9,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554148","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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