Pub Date : 2023-11-16DOI: 10.1142/s0129054123500235
Fujita Satoshi
In this paper, we consider the problem of covering the vertex set of a given graph by [Formula: see text] trees so as to minimize the maximum weight of the trees, as a subproblem of the multi-ferry scheduling problem proposed by Zhao and Ammar. After pointing out that the approximation ratio of a greedy scheme based on the Kruskal’s algorithm is provably bad, we show that the approximation ratio cannot be better than 3/2 for [Formula: see text] even when the edge selection criterion is modified so as to minimize the increase in the maximum weight in the collection of trees. We then propose two polynomial-time algorithms with a guaranteed approximation ratio. The first algorithm achieves 3-approximation for the class of graphs in which the edge weights satisfy the triangle inequality. The second algorithm achieves 4-approximation for any connected graph with arbitrary edge weights.
{"title":"Approximating Minimum k-Tree Cover of a Connected Graph Inspired by the Multi-Ferry Routing in Delay Tolerant Networks","authors":"Fujita Satoshi","doi":"10.1142/s0129054123500235","DOIUrl":"https://doi.org/10.1142/s0129054123500235","url":null,"abstract":"In this paper, we consider the problem of covering the vertex set of a given graph by [Formula: see text] trees so as to minimize the maximum weight of the trees, as a subproblem of the multi-ferry scheduling problem proposed by Zhao and Ammar. After pointing out that the approximation ratio of a greedy scheme based on the Kruskal’s algorithm is provably bad, we show that the approximation ratio cannot be better than 3/2 for [Formula: see text] even when the edge selection criterion is modified so as to minimize the increase in the maximum weight in the collection of trees. We then propose two polynomial-time algorithms with a guaranteed approximation ratio. The first algorithm achieves 3-approximation for the class of graphs in which the edge weights satisfy the triangle inequality. The second algorithm achieves 4-approximation for any connected graph with arbitrary edge weights.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139270076","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 : 2023-11-16DOI: 10.1142/s0129054123480088
Manfred Droste, Gustav Grabolle, George Rahonis
We introduce a weighted linear dynamic logic (weighted LDL for short) and show the expressive equivalence of its formulas to weighted rational expressions. This adds a new characterization for recognizable series to the fundamental Schützenberger theorem. Surprisingly, the equivalence does not require any restriction to our weighted LDL. Our results hold over arbitrary (resp. totally complete) semirings for finite (resp. infinite) words. As a consequence, the equivalence problem for weighted LDL formulas over fields is decidable in doubly exponential time. In contrast to classical logics, we show that our weighted LDL is expressively incomparable to weighted LTL for finite words. We determine a fragment of the weighted LTL such that series over finite and infinite words definable by LTL formulas in this fragment are definable also by weighted LDL formulas. This is an extended version of [17].
{"title":"Weighted Linear Dynamic Logic","authors":"Manfred Droste, Gustav Grabolle, George Rahonis","doi":"10.1142/s0129054123480088","DOIUrl":"https://doi.org/10.1142/s0129054123480088","url":null,"abstract":"<p>We introduce a weighted linear dynamic logic (weighted LDL for short) and show the expressive equivalence of its formulas to weighted rational expressions. This adds a new characterization for recognizable series to the fundamental Schützenberger theorem. Surprisingly, the equivalence does not require any restriction to our weighted LDL. Our results hold over arbitrary (resp. totally complete) semirings for finite (resp. infinite) words. As a consequence, the equivalence problem for weighted LDL formulas over fields is decidable in doubly exponential time. In contrast to classical logics, we show that our weighted LDL is expressively incomparable to weighted LTL for finite words. We determine a fragment of the weighted LTL such that series over finite and infinite words definable by LTL formulas in this fragment are definable also by weighted LDL formulas. This is an extended version of [17].</p>","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140075369","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 : 2023-11-15DOI: 10.1142/s012905412348009x
Emmanuel Filiot, Sarah Winter
The synthesis problem asks to automatically generate, if it exists, an algorithm from a specification of correct input-output pairs. In this paper, we consider the synthesis of computable functions of infinite words, for a classical Turing computability notion over infinite inputs. We consider specifications which are rational relations of infinite words, i.e., specifications defined by non-deterministic parity transducers. We prove that the synthesis problem of computable functions from rational specifications is undecidable. We provide an incomplete but sound reduction to some parity game, such that if Eve wins the game, then the rational specification is realizable by a computable function. We prove that this function is even computable by a deterministic two-way transducer. We provide a sufficient condition under which the latter game reduction is complete. This entails the decidability of the synthesis problem of computable functions, which we proved to be ExpTime-complete, for a large subclass of rational specifications, namely deterministic rational specifications. This subclass contains the class of automatic relations over infinite words, a yardstick in reactive synthesis.
{"title":"Synthesizing Computable Functions from Rational Specifications Over Infinite Words","authors":"Emmanuel Filiot, Sarah Winter","doi":"10.1142/s012905412348009x","DOIUrl":"https://doi.org/10.1142/s012905412348009x","url":null,"abstract":"The synthesis problem asks to automatically generate, if it exists, an algorithm from a specification of correct input-output pairs. In this paper, we consider the synthesis of computable functions of infinite words, for a classical Turing computability notion over infinite inputs. We consider specifications which are rational relations of infinite words, i.e., specifications defined by non-deterministic parity transducers. We prove that the synthesis problem of computable functions from rational specifications is undecidable. We provide an incomplete but sound reduction to some parity game, such that if Eve wins the game, then the rational specification is realizable by a computable function. We prove that this function is even computable by a deterministic two-way transducer. We provide a sufficient condition under which the latter game reduction is complete. This entails the decidability of the synthesis problem of computable functions, which we proved to be ExpTime-complete, for a large subclass of rational specifications, namely deterministic rational specifications. This subclass contains the class of automatic relations over infinite words, a yardstick in reactive synthesis.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227556","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 : 2023-11-11DOI: 10.1142/s0129054123500272
Hong Gao, Yunlei Zhang, Yuqi Wang, Yuanyuan Guo, Xing Liu, Renbang Liu, Changqing Xi, Yuansheng Yang
Let [Formula: see text] be a graph and [Formula: see text] be an integer representing [Formula: see text] colors. There is a function [Formula: see text] from [Formula: see text] to the power set of [Formula: see text] colors satisfying every vertex [Formula: see text] assigned [Formula: see text] under [Formula: see text] in its neighborhood has all the colors, then [Formula: see text] is called a [Formula: see text]-rainbow dominating function ([Formula: see text]RDF) on [Formula: see text]. The weight of [Formula: see text] is the sum of the number of colors on each vertex all over the graph. The [Formula: see text]-rainbow domination number of [Formula: see text] is the minimum weight of [Formula: see text]RDFs on [Formula: see text], denoted by [Formula: see text]. The aim of this paper is to investigate the [Formula: see text]-rainbow ([Formula: see text]) domination number of the Cartesian product of paths [Formula: see text] and the Cartesian product of paths and cycles [Formula: see text]. For [Formula: see text], we determine the value [Formula: see text] and present [Formula: see text] for [Formula: see text]. For [Formula: see text], we determine the values of [Formula: see text] for [Formula: see text] or [Formula: see text] and [Formula: see text] for [Formula: see text] or [Formula: see text].
设[公式:见文本]为图形，[公式:见文本]为表示[公式:见文本]颜色的整数。有一个函数[Formula: see text]从[Formula: see text]到[Formula: see text]颜色的幂集满足在[Formula: see text]下分配的[Formula: see text]的每个顶点[Formula: see text]在它的邻域中具有所有的颜色，那么[Formula: see text]就被称为[Formula: see text]上的[Formula: see text]-彩虹支配函数([Formula: see text]RDF)。[公式:见文本]的权重是图形上每个顶点的颜色数量之和。[公式:见文]的[公式:见文]-彩虹支配数是[公式:见文]上[公式:见文]rdf的最小权值，用[公式:见文]表示。本文的目的是研究路径的笛卡尔积[公式:见文]和路径与循环的笛卡尔积[公式:见文]的[公式:见文]-彩虹([公式:见文])的支配数。对于[Formula: see text]，我们确定值[Formula: see text]，并为[Formula: see text]呈现[Formula: see text]。对于[公式:见文本]，我们为[公式:见文本]或[公式:见文本]确定[公式:见文本]的值，为[公式:见文本]或[公式:见文本]确定[公式:见文本]的值。
{"title":"Rainbow Domination in Cartesian Product of Paths and Cycles","authors":"Hong Gao, Yunlei Zhang, Yuqi Wang, Yuanyuan Guo, Xing Liu, Renbang Liu, Changqing Xi, Yuansheng Yang","doi":"10.1142/s0129054123500272","DOIUrl":"https://doi.org/10.1142/s0129054123500272","url":null,"abstract":"Let [Formula: see text] be a graph and [Formula: see text] be an integer representing [Formula: see text] colors. There is a function [Formula: see text] from [Formula: see text] to the power set of [Formula: see text] colors satisfying every vertex [Formula: see text] assigned [Formula: see text] under [Formula: see text] in its neighborhood has all the colors, then [Formula: see text] is called a [Formula: see text]-rainbow dominating function ([Formula: see text]RDF) on [Formula: see text]. The weight of [Formula: see text] is the sum of the number of colors on each vertex all over the graph. The [Formula: see text]-rainbow domination number of [Formula: see text] is the minimum weight of [Formula: see text]RDFs on [Formula: see text], denoted by [Formula: see text]. The aim of this paper is to investigate the [Formula: see text]-rainbow ([Formula: see text]) domination number of the Cartesian product of paths [Formula: see text] and the Cartesian product of paths and cycles [Formula: see text]. For [Formula: see text], we determine the value [Formula: see text] and present [Formula: see text] for [Formula: see text]. For [Formula: see text], we determine the values of [Formula: see text] for [Formula: see text] or [Formula: see text] and [Formula: see text] for [Formula: see text] or [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087194","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 : 2023-11-10DOI: 10.1142/s0129054123500259
Mirosław Kowaluk, Andrzej Lingas
The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for [Formula: see text] Boolean matrices of the form [Formula: see text] has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the classical computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time [Formula: see text], where [Formula: see text] satisfies the equation [Formula: see text] and [Formula: see text] is the exponent of the multiplication of an [Formula: see text] matrix by an [Formula: see text] matrix. Next, we consider a relaxed version of the MW problem (in the classical model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most [Formula: see text] in time [Formula: see text] Then, by reducing the relaxed problem to the so called [Formula: see text]-witness problem, we provide an algorithm that reports for each non-zero entry [Formula: see text] of the product matrix [Formula: see text] a witness of rank [Formula: see text], where [Formula: see text] is the number of witnesses for [Formula: see text], with high probability. The algorithm runs in [Formula: see text] time, where [Formula: see text].
{"title":"Quantum and Approximation Algorithms for Maximum Witnesses of Boolean Matrix Products","authors":"Mirosław Kowaluk, Andrzej Lingas","doi":"10.1142/s0129054123500259","DOIUrl":"https://doi.org/10.1142/s0129054123500259","url":null,"abstract":"The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for [Formula: see text] Boolean matrices of the form [Formula: see text] has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the classical computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time [Formula: see text], where [Formula: see text] satisfies the equation [Formula: see text] and [Formula: see text] is the exponent of the multiplication of an [Formula: see text] matrix by an [Formula: see text] matrix. Next, we consider a relaxed version of the MW problem (in the classical model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most [Formula: see text] in time [Formula: see text] Then, by reducing the relaxed problem to the so called [Formula: see text]-witness problem, we provide an algorithm that reports for each non-zero entry [Formula: see text] of the product matrix [Formula: see text] a witness of rank [Formula: see text], where [Formula: see text] is the number of witnesses for [Formula: see text], with high probability. The algorithm runs in [Formula: see text] time, where [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087720","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 : 2023-11-01DOI: 10.1142/s0129054123430025
Stefan Hoffmann
We investigate the state complexity of the permutation operation, or the commutative closure, on Alphabetical Pattern Constraints (APCs). This class corresponds to level [Formula: see text] of the Straubing-Thérien hierarchy and includes the finite, the piecewise testable, or [Formula: see text]-trivial, and the [Formula: see text]-trivial and [Formula: see text]-trivial languages. We give a sharp state complexity bound expressed in terms of the longest strings in the unary projection languages of an associated finite language. Additionally, for a subclass, we give sharp bounds expressed in terms of the size of a recognizing input automaton and the size of the alphabet. We also state a related state complexity bound for the commutative closure on finite languages. Lastly, we investigate the language inclusion, equivalence and universality problems on APCs up to permutational, or Parikh, equivalence. These problems are known to be [Formula: see text]-complete on APCs in general, even for fixed alphabets. We show them to be decidable in polynomial time for fixed alphabets if we only want to solve them up to Parikh equivalence. We also correct a mistake from the conference version in a bound on the size of recognizing automata for the commutative closure.
{"title":"State Complexity of Permutation and the Language Inclusion Problem up to Parikh Equivalence on Alphabetical Pattern Constraints and Partially Ordered NFAs","authors":"Stefan Hoffmann","doi":"10.1142/s0129054123430025","DOIUrl":"https://doi.org/10.1142/s0129054123430025","url":null,"abstract":"We investigate the state complexity of the permutation operation, or the commutative closure, on Alphabetical Pattern Constraints (APCs). This class corresponds to level [Formula: see text] of the Straubing-Thérien hierarchy and includes the finite, the piecewise testable, or [Formula: see text]-trivial, and the [Formula: see text]-trivial and [Formula: see text]-trivial languages. We give a sharp state complexity bound expressed in terms of the longest strings in the unary projection languages of an associated finite language. Additionally, for a subclass, we give sharp bounds expressed in terms of the size of a recognizing input automaton and the size of the alphabet. We also state a related state complexity bound for the commutative closure on finite languages. Lastly, we investigate the language inclusion, equivalence and universality problems on APCs up to permutational, or Parikh, equivalence. These problems are known to be [Formula: see text]-complete on APCs in general, even for fixed alphabets. We show them to be decidable in polynomial time for fixed alphabets if we only want to solve them up to Parikh equivalence. We also correct a mistake from the conference version in a bound on the size of recognizing automata for the commutative closure.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135371911","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 : 2023-10-28DOI: 10.1142/s0129054123500260
Manjay Kumar, P. Venkata Subba Reddy
For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.
对于没有孤立顶点的简单无向图[公式:见文]，满足以下两个条件的函数[公式:见文]称为[公式:见文]的总2-彩虹支配函数(T2RDF)。(i)对于所有[公式:见文]，如果[公式:见文]，则[公式:见文];(ii)每个[公式:见文]与[公式:见文]的顶点[公式:见文]相邻。[Formula: see text]的T2RDF [Formula: see text]的权重是值[Formula: see text]。总2彩虹控制数是一个T2RDF在[公式:见文本]上的最小权重，用[公式:见文本]表示。最小总2彩虹支配问题(MT2RDP)是在输入图中找到一个最小权重的T2RDF。在本文中，我们证明了判定星形凸二部图、梳状凸二部图、分裂图和平面图的[公式:见文]是否有最大权值的T2RDF的问题是np完全的。在积极的方面，我们证明了MT2RDP对于阈值图、链图和有界树宽度图是线性时间可解的。从近似的角度来看，我们表明MT2RDP不能在[公式:见文]内近似任何[公式:见文]，除非[公式:见文]，并提出[公式:见文]-近似算法。进一步，我们证明了MT2RDP对于最大度为4的图是apx完全的。其次，证明了控制问题和总2彩虹控制问题在计算复杂度方面是不等价的。最后，给出了MT2RDP的整数线性规划公式。
{"title":"Total 2-Rainbow Domination in Graphs: Complexity and Algorithms","authors":"Manjay Kumar, P. Venkata Subba Reddy","doi":"10.1142/s0129054123500260","DOIUrl":"https://doi.org/10.1142/s0129054123500260","url":null,"abstract":"For a simple, undirected graph [Formula: see text] without isolated vertices, a function [Formula: see text] which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of [Formula: see text]. (i) For all [Formula: see text], if [Formula: see text] then [Formula: see text] and (ii) Every [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. The weight of a T2RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. The total 2-rainbow domination number is the minimum weight of a T2RDF on [Formula: see text] and is denoted by [Formula: see text]. The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if [Formula: see text] has a T2RDF of weight at most [Formula: see text] for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text] and also propose [Formula: see text]-approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232567","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 : 2023-10-19DOI: 10.1142/s0129054123500247
H. Naresh Kumar, Mustapha Chellali, Y. B. Venkatakrishnan
A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.
{"title":"Algorithmic Aspects of Total Vertex-Edge Domination in Graphs","authors":"H. Naresh Kumar, Mustapha Chellali, Y. B. Venkatakrishnan","doi":"10.1142/s0129054123500247","DOIUrl":"https://doi.org/10.1142/s0129054123500247","url":null,"abstract":"A vertex [Formula: see text] of a simple graph [Formula: see text] ve-dominates every edge incident to [Formula: see text] as well as every edge adjacent to these incident edges. A set [Formula: see text] is a total vertex-edge dominating set if every edge of [Formula: see text] is ve-dominated by a vertex of [Formula: see text] and the subgraph induced by [Formula: see text] has no isolated vertex. The total vertex-edge domination problem is to find a total vertex-edge dominating set of minimum cardinality. In this paper, we first show that the total vertex-edge domination problem is NP-complete for chordal graphs. Then we provide a linear-time algorithm for this problem in trees. Moreover, we show that the minimum total vertex-edge domination problem cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text]). Finally, we prove that the minimum total vertex-edge domination problem is APX-complete for bounded-degree graphs.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730676","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 : 2023-10-19DOI: 10.1142/s0129054123500223
Zengtai Gong, Lele He
The connectivity is one of the crucial parameters of network used to transport network flow, routing problems and bandwidth allocation problems. The increased connectivity makes a network more stable. In this paper, a new parameter called connectivity status of a vertex is introduced in the intuitionistic fuzzy graph. The related definitions and propositions of connectivity status of a vertex are proposed and investigated in an intuitionistic fuzzy graph. Specifically, connectivity status and status sequence are defined and analysed with various examples. After deleting a vertex, we classify the vertices of an intuitionistic fuzzy graph as connectivity status enhancing vertices, connectivity status neutral vertices and connectivity status reducing vertices because of the change of connectivity status. Finally, we establish two algorithms for these concepts and give an application to illustrate feasibility of algorithms.
{"title":"Connectivity Status of Intuitionistic Fuzzy Graphs and Its Applications","authors":"Zengtai Gong, Lele He","doi":"10.1142/s0129054123500223","DOIUrl":"https://doi.org/10.1142/s0129054123500223","url":null,"abstract":"The connectivity is one of the crucial parameters of network used to transport network flow, routing problems and bandwidth allocation problems. The increased connectivity makes a network more stable. In this paper, a new parameter called connectivity status of a vertex is introduced in the intuitionistic fuzzy graph. The related definitions and propositions of connectivity status of a vertex are proposed and investigated in an intuitionistic fuzzy graph. Specifically, connectivity status and status sequence are defined and analysed with various examples. After deleting a vertex, we classify the vertices of an intuitionistic fuzzy graph as connectivity status enhancing vertices, connectivity status neutral vertices and connectivity status reducing vertices because of the change of connectivity status. Finally, we establish two algorithms for these concepts and give an application to illustrate feasibility of algorithms.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135730644","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 : 2023-10-18DOI: 10.1142/s0129054123470044
Bogdan Aman
A distributed reaction system models a system composed of several reaction systems. Each reaction system has its own set of reactions, while the background set is the same for all reaction systems. At each transition of the distributed reaction system, the environment provides an arbitrary context containing symbols for each reaction system and also it specifies which reaction systems are active. On the other hand, a distributed communicating reaction system with direct communication models a system composed of several reaction systems that are able to communicate products or reactions, while the environment provides a context similar to that for distributed reaction systems. In this paper, it is proved that these distributed variants of reaction systems can be related by establishing translations of distributed reaction systems into distributed communicating reaction systems with direct communication and the other way round.
{"title":"Relating Various Types of Distributed Reaction Systems","authors":"Bogdan Aman","doi":"10.1142/s0129054123470044","DOIUrl":"https://doi.org/10.1142/s0129054123470044","url":null,"abstract":"A distributed reaction system models a system composed of several reaction systems. Each reaction system has its own set of reactions, while the background set is the same for all reaction systems. At each transition of the distributed reaction system, the environment provides an arbitrary context containing symbols for each reaction system and also it specifies which reaction systems are active. On the other hand, a distributed communicating reaction system with direct communication models a system composed of several reaction systems that are able to communicate products or reactions, while the environment provides a context similar to that for distributed reaction systems. In this paper, it is proved that these distributed variants of reaction systems can be related by establishing translations of distributed reaction systems into distributed communicating reaction systems with direct communication and the other way round.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135883739","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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