Pub Date : 2023-05-04DOI: 10.1142/s0129054123450016
Viktor Olejár, Alexander Szabari
A class of languages is closed under a given operation if the resulting language belongs to this class whenever the operands belong to it. We examine the closure properties of various subclasses of regular languages under basic operations of intersection, union, concatenation and power, positive closure and star, reversal, and complementation. We consider the following classes: definite languages and their variants (left ideal, finitely generated left ideal, symmetric definite, generalized definite and combinational), two-sided comets and their variants comets and stars, and the classes of singleton, finite, ordered, star-free, and power-separating languages. We also give an overview about subclasses of convex languages (classes of ideal, free, and closed languages), union-free languages, and group languages. We summarize some inclusion relations between these classes. Subsequently, for all pairs of a class and an operation, we provide an answer whether this class is closed under this operation or not.
{"title":"Closure Properties of Subregular Languages Under Operations","authors":"Viktor Olejár, Alexander Szabari","doi":"10.1142/s0129054123450016","DOIUrl":"https://doi.org/10.1142/s0129054123450016","url":null,"abstract":"A class of languages is closed under a given operation if the resulting language belongs to this class whenever the operands belong to it. We examine the closure properties of various subclasses of regular languages under basic operations of intersection, union, concatenation and power, positive closure and star, reversal, and complementation. We consider the following classes: definite languages and their variants (left ideal, finitely generated left ideal, symmetric definite, generalized definite and combinational), two-sided comets and their variants comets and stars, and the classes of singleton, finite, ordered, star-free, and power-separating languages. We also give an overview about subclasses of convex languages (classes of ideal, free, and closed languages), union-free languages, and group languages. We summarize some inclusion relations between these classes. Subsequently, for all pairs of a class and an operation, we provide an answer whether this class is closed under this operation or not.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64087728","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 : 2023-04-27DOI: 10.1142/s0129054123410046
Raffael M. Paranhos, Janio Carlos Nascimento Silva, U. Souza
The main complexity classes of the Parameterized Intractability Theory are based on weighted Boolean circuit satisfiability problems and organized into a hierarchy so-called W-hierarchy. The W-hierarchy enables fine-grained complexity analyses of parameterized problems that are unlikely to belong to the FPT class. In this paper, we introduce the Th-hierarchy, a natural generalization of the W-hierarchy defined by unweighted threshold circuit satisfiability problems. Investigating the relationship between Th-hierarchy and W-hierarchy, we discuss the complexity of transforming Threshold circuits into Boolean circuits, and observe that sorting networks are powerful tools to handle such transformations. First, we show that these hierarchies collapse at the last level (W[P][Formula: see text][Formula: see text][Formula: see text]Th[P]). After that, we present a time complexity analysis of an AKS sorting network construction, which supports some of our results. Finally, we prove that Th[[Formula: see text]] [Formula: see text] W[SAT] for every [Formula: see text]. As a by-product, our studies suggest that it is relevant to consider a new class based on logarithmic depth circuits in the W-hierarchy.
{"title":"Parameterized Complexity Classes Defined by Threshold Circuits and Their Connection with Sorting Networks","authors":"Raffael M. Paranhos, Janio Carlos Nascimento Silva, U. Souza","doi":"10.1142/s0129054123410046","DOIUrl":"https://doi.org/10.1142/s0129054123410046","url":null,"abstract":"The main complexity classes of the Parameterized Intractability Theory are based on weighted Boolean circuit satisfiability problems and organized into a hierarchy so-called W-hierarchy. The W-hierarchy enables fine-grained complexity analyses of parameterized problems that are unlikely to belong to the FPT class. In this paper, we introduce the Th-hierarchy, a natural generalization of the W-hierarchy defined by unweighted threshold circuit satisfiability problems. Investigating the relationship between Th-hierarchy and W-hierarchy, we discuss the complexity of transforming Threshold circuits into Boolean circuits, and observe that sorting networks are powerful tools to handle such transformations. First, we show that these hierarchies collapse at the last level (W[P][Formula: see text][Formula: see text][Formula: see text]Th[P]). After that, we present a time complexity analysis of an AKS sorting network construction, which supports some of our results. Finally, we prove that Th[[Formula: see text]] [Formula: see text] W[SAT] for every [Formula: see text]. As a by-product, our studies suggest that it is relevant to consider a new class based on logarithmic depth circuits in the W-hierarchy.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43504068","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 : 2023-04-27DOI: 10.1142/s0129054123410010
Sai Ji, Yukun Cheng, Jingjing Tan, Zhongrui Zhao
Correlation clustering problem (CorCP) is a classical clustering problem, which clusters data based on the similarity of data set, and has many applications in interaction networks, cross-lingual link detection, and communication networks, etc. In this paper, we study a practical generalization of the CorCP, called the capacitated correlation clustering problem (the capacitated CorCP), by constructing a labeled complete graph. On this labeled complete graph, each vertex represents a piece of data. If two pieces of data are similar, then the edge between the corresponding vertices is marked by a positive label [Formula: see text]. Otherwise, this edge is marked by a negative label −. The objective of the capacitated CorCP is to group some similar data sets into one cluster as far as possible, while satisfying the cluster capacity constraint. To achieve this objective, we shall partition the vertex set of the labeled complete graph into several clusters, each cluster’s size subjecting to an upper bound, so as to minimize the number of disagreements. Here the number of disagreements is defined as the total number of the edges with positive labels between clusters and the edges with negative labels within clusters. Different with the previous algorithm in [18], which subjects to the constraint on the cluster size by a penalty measure, we design an algorithm for the capacitated CorCP to directly output a feasible solution by iteratively constructing clusters based on a preset threshold. Through carefully setting the threshold and sophisticatedly analyzing, our algorithm is proved to have an improved approximation ratio of 5.37. In addition, we also conduct a series of numerical experiments to demonstrate the effectiveness of our algorithm.
{"title":"An Improved Approximation Algorithm for the Capacitated Correlation Clustering Problem","authors":"Sai Ji, Yukun Cheng, Jingjing Tan, Zhongrui Zhao","doi":"10.1142/s0129054123410010","DOIUrl":"https://doi.org/10.1142/s0129054123410010","url":null,"abstract":"Correlation clustering problem (CorCP) is a classical clustering problem, which clusters data based on the similarity of data set, and has many applications in interaction networks, cross-lingual link detection, and communication networks, etc. In this paper, we study a practical generalization of the CorCP, called the capacitated correlation clustering problem (the capacitated CorCP), by constructing a labeled complete graph. On this labeled complete graph, each vertex represents a piece of data. If two pieces of data are similar, then the edge between the corresponding vertices is marked by a positive label [Formula: see text]. Otherwise, this edge is marked by a negative label −. The objective of the capacitated CorCP is to group some similar data sets into one cluster as far as possible, while satisfying the cluster capacity constraint. To achieve this objective, we shall partition the vertex set of the labeled complete graph into several clusters, each cluster’s size subjecting to an upper bound, so as to minimize the number of disagreements. Here the number of disagreements is defined as the total number of the edges with positive labels between clusters and the edges with negative labels within clusters. Different with the previous algorithm in [18], which subjects to the constraint on the cluster size by a penalty measure, we design an algorithm for the capacitated CorCP to directly output a feasible solution by iteratively constructing clusters based on a preset threshold. Through carefully setting the threshold and sophisticatedly analyzing, our algorithm is proved to have an improved approximation ratio of 5.37. In addition, we also conduct a series of numerical experiments to demonstrate the effectiveness of our algorithm.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119299","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 : 2023-04-21DOI: 10.1142/s012905412341006x
Sankardeep Chakraborty, Seungbum Jo, K. Sadakane, Srinivasa Rao Satti
We design succinct encodings of series-parallel, block-cactus and 3-leaf power graphs while supporting the basic navigational queries such as degree, adjacency and neighborhood optimally in the RAM model with logarithmic word size. One salient feature of our representation is that it can achieve optimal space even though the exact space lower bound for these graph classes is not known. For these graph classes, we provide succinct data structures with optimal query support for the first time in the literature. For series-parallel multigraphs, our work also extends the works of Uno et al. (Disc. Math. Alg. and Appl., 2013) and Blelloch and Farzan (CPM, 2010) to produce optimal bounds.
{"title":"Succinct Data Structures for SP, Block-Cactus and 3-Leaf Power Graphs","authors":"Sankardeep Chakraborty, Seungbum Jo, K. Sadakane, Srinivasa Rao Satti","doi":"10.1142/s012905412341006x","DOIUrl":"https://doi.org/10.1142/s012905412341006x","url":null,"abstract":"We design succinct encodings of series-parallel, block-cactus and 3-leaf power graphs while supporting the basic navigational queries such as degree, adjacency and neighborhood optimally in the RAM model with logarithmic word size. One salient feature of our representation is that it can achieve optimal space even though the exact space lower bound for these graph classes is not known. For these graph classes, we provide succinct data structures with optimal query support for the first time in the literature. For series-parallel multigraphs, our work also extends the works of Uno et al. (Disc. Math. Alg. and Appl., 2013) and Blelloch and Farzan (CPM, 2010) to produce optimal bounds.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47111268","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 : 2023-04-21DOI: 10.1142/s0129054123410034
R. Inkulu, Pawan Kumar
Given a set [Formula: see text] of [Formula: see text] pairwise disjoint convex polygonal obstacles in the plane, defined with [Formula: see text] vertices, we preprocess [Formula: see text] and compute one routing table at each vertex in a subset of vertices of [Formula: see text]. For routing a packet from any vertex [Formula: see text] to any vertex [Formula: see text], our scheme computes a routing path with a multiplicative stretch [Formula: see text] and an additive stretch [Formula: see text], by consulting routing tables at only a subset of vertices along that path. Here, [Formula: see text] is the number of obstacles of [Formula: see text] the routing path intersects, and [Formula: see text] depends on the geometry of obstacles in [Formula: see text]. During the preprocessing phase, we construct routing tables of size [Formula: see text] in [Formula: see text] time, where [Formula: see text] is an input parameter.
给定平面上由[公式:见文]顶点定义的[公式:见文]对不相交凸多边形障碍物的集合[公式:见文]，我们对[公式:见文]顶点子集中的[公式:见文]顶点进行预处理，并在每个顶点计算一个路由表。对于从任意顶点[公式:见文本]路由数据包到任意顶点[公式:见文本]，我们的方案计算一个具有乘法拉伸[公式:见文本]和加法拉伸[公式:见文本]的路由路径，通过仅在该路径上的一个子集的顶点上查询路由表。其中，[公式:见文]为路径所交叉的[公式:见文]障碍物的数量，[公式:见文]取决于[公式:见文]中障碍物的几何形状。在预处理阶段，我们在[Formula: see text] time中构造大小为[Formula: see text]的路由表，其中[Formula: see text]为输入参数。
{"title":"Routing Among Convex Polygonal Obstacles in the Plane","authors":"R. Inkulu, Pawan Kumar","doi":"10.1142/s0129054123410034","DOIUrl":"https://doi.org/10.1142/s0129054123410034","url":null,"abstract":"Given a set [Formula: see text] of [Formula: see text] pairwise disjoint convex polygonal obstacles in the plane, defined with [Formula: see text] vertices, we preprocess [Formula: see text] and compute one routing table at each vertex in a subset of vertices of [Formula: see text]. For routing a packet from any vertex [Formula: see text] to any vertex [Formula: see text], our scheme computes a routing path with a multiplicative stretch [Formula: see text] and an additive stretch [Formula: see text], by consulting routing tables at only a subset of vertices along that path. Here, [Formula: see text] is the number of obstacles of [Formula: see text] the routing path intersects, and [Formula: see text] depends on the geometry of obstacles in [Formula: see text]. During the preprocessing phase, we construct routing tables of size [Formula: see text] in [Formula: see text] time, where [Formula: see text] is an input parameter.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":"82 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135518303","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 : 2023-04-11DOI: 10.1142/s0129054123500041
Processor fault diagnosis is intended to identify faulty processors of a multicomputer system, guaranteeing its high reliability and availability. The [Formula: see text]-good-neighbor conditional diagnosability is a novel fault diagnosis method for various networks. Under the PMC model and comparison model, the [Formula: see text]-good-neighbor conditional diagnosabilities of hypermesh optical interconnection networks are determined, respectively. Directly applying the results, the [Formula: see text]-good-neighbor conditional diagnosabilities of hypercubes are derived under the PMC and comparison models.
{"title":"The g-Good-Neighbor Conditional Diagnosabilities of Hypermesh Optical Interconnection Networks Under the PMC and Comparison Models","authors":"","doi":"10.1142/s0129054123500041","DOIUrl":"https://doi.org/10.1142/s0129054123500041","url":null,"abstract":"Processor fault diagnosis is intended to identify faulty processors of a multicomputer system, guaranteeing its high reliability and availability. The [Formula: see text]-good-neighbor conditional diagnosability is a novel fault diagnosis method for various networks. Under the PMC model and comparison model, the [Formula: see text]-good-neighbor conditional diagnosabilities of hypermesh optical interconnection networks are determined, respectively. Directly applying the results, the [Formula: see text]-good-neighbor conditional diagnosabilities of hypercubes are derived under the PMC and comparison models.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45339284","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 : 2023-03-29DOI: 10.1142/s0129054123500065
Fatemeh Keshavarz-Kohjerdi, A. Bagheri
One of the well-known NP-hard optimization problems in graph theory is finding the longest path in a graph. This problem remains NP-hard for general grid graphs, and its complexity is open for grid graphs that have a limited number of holes. In this paper, we study this problem for odd-sized [Formula: see text]-shaped grid graphs, i.e. a rectangular grid graph with a rectangular hole. We show that this problem can be solved in linear time.
{"title":"The Longest Path Problem in Odd-Sized O-Shaped Grid Graphs","authors":"Fatemeh Keshavarz-Kohjerdi, A. Bagheri","doi":"10.1142/s0129054123500065","DOIUrl":"https://doi.org/10.1142/s0129054123500065","url":null,"abstract":"One of the well-known NP-hard optimization problems in graph theory is finding the longest path in a graph. This problem remains NP-hard for general grid graphs, and its complexity is open for grid graphs that have a limited number of holes. In this paper, we study this problem for odd-sized [Formula: see text]-shaped grid graphs, i.e. a rectangular grid graph with a rectangular hole. We show that this problem can be solved in linear time.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45745611","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 : 2023-03-28DOI: 10.1142/s0129054123500053
Ruyan Guo, Yan Wang, Jianxi Fan, Weibei Fan
In recent years, the growth of data has promoted the development of parallel and distributed systems. Graph embedding is of great importance in improving parallel and distributed system performance. The quality of an embedding can be measured by many important metrics, and wirelength is one of the critical metrics related to communication performance and layout costs of physical systems. The hierarchical cubic network is a well-performing interconnection network and the [Formula: see text]-rooted complete binary tree is a data structure with a hierarchical relationship among its various elements in algorithms and programming. In this paper, we solve the problem of the embedding of hierarchical cubic networks into [Formula: see text]-rooted complete binary trees with minimum wirelength. We first study the optimal set of the hierarchical cubic network, and propose algorithms to give embedding [Formula: see text] which is an embedding scheme of hierarchical cubic networks into [Formula: see text]-rooted complete binary trees with minimum wirelength. Moreover, we give the exact minimum wirelength of this embedding. Finally, we conduct comparative experiments to evaluate the performance of embedding [Formula: see text].
{"title":"Embedding Hierarchical Cubic Networks into k-Rooted Complete Binary Trees for Minimum Wirelength","authors":"Ruyan Guo, Yan Wang, Jianxi Fan, Weibei Fan","doi":"10.1142/s0129054123500053","DOIUrl":"https://doi.org/10.1142/s0129054123500053","url":null,"abstract":"In recent years, the growth of data has promoted the development of parallel and distributed systems. Graph embedding is of great importance in improving parallel and distributed system performance. The quality of an embedding can be measured by many important metrics, and wirelength is one of the critical metrics related to communication performance and layout costs of physical systems. The hierarchical cubic network is a well-performing interconnection network and the [Formula: see text]-rooted complete binary tree is a data structure with a hierarchical relationship among its various elements in algorithms and programming. In this paper, we solve the problem of the embedding of hierarchical cubic networks into [Formula: see text]-rooted complete binary trees with minimum wirelength. We first study the optimal set of the hierarchical cubic network, and propose algorithms to give embedding [Formula: see text] which is an embedding scheme of hierarchical cubic networks into [Formula: see text]-rooted complete binary trees with minimum wirelength. Moreover, we give the exact minimum wirelength of this embedding. Finally, we conduct comparative experiments to evaluate the performance of embedding [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43895964","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 : 2023-03-22DOI: 10.1142/s0129054123460012
Xiaofei Liu, Weidong Li
Given a graph [Formula: see text], a set of [Formula: see text] source-sink pairs [Formula: see text] [Formula: see text] and a profit bound [Formula: see text], every edge [Formula: see text] has a cost [Formula: see text], and every source-sink pair [Formula: see text] has a profit [Formula: see text] and a penalty [Formula: see text]. The [Formula: see text]-prize-collecting multicut problem ([Formula: see text]-PCMP) is to find a multicut [Formula: see text] such that the objective cost, which consists of the total cost of the edges in [Formula: see text] and the total penalty of the pairs still connected after removing [Formula: see text], is minimized and the total profit of the disconnected pairs by removing [Formula: see text] is at least [Formula: see text]. In this paper, we firstly consider the [Formula: see text]-PCMP in paths, and prove that it is [Formula: see text]-hard even when [Formula: see text] for any [Formula: see text]. Then, we present a fully polynomial time approximation scheme (FPTAS) whose running time is [Formula: see text] for the [Formula: see text]-PCMP in paths. Based on this algorithm, we present an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in spider graphs, and an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in rings, respectively, where [Formula: see text] is the number of leaves of spider graph.
给定一个图[公式:见文]，一组[公式:见文]源汇对[公式:见文][公式:见文]和一个利润界限[公式:见文]，每条边[公式:见文]都有一个成本[公式:见文]，每个源汇对[公式:见文]都有一个利润[公式:见文]和一个损失[公式:见文]。[公式:见文]-计奖多切口问题([公式:见文]-PCMP)是寻找一个多切口[公式:见文]，使[公式:见文]中边缘的总成本和去除[公式:见文]后仍然连接的对的总惩罚的客观成本最小，并且通过去除[公式:见文]而断开的对的总利润至少为[公式:见文]。本文首先考虑路径中的[Formula: see text]-PCMP，并证明对于任何[Formula: see text]，即使[Formula: see text]是[Formula: see text]-hard。然后，我们提出了一个完全多项式时间近似方案(FPTAS)，其运行时间为[公式:见文本]-路径中的pcmp。在此算法的基础上，我们提出了一个运行时间为[公式:见文]的蜘蛛图-PCMP的FPTAS，以及一个运行时间为[公式:见文]的[公式:见文]的[圆环]-PCMP的FPTAS，其中[公式:见文]为蜘蛛图的叶子数。
{"title":"The B-Prize-Collecting Multicut Problem in Paths, Spider Graphs and Rings","authors":"Xiaofei Liu, Weidong Li","doi":"10.1142/s0129054123460012","DOIUrl":"https://doi.org/10.1142/s0129054123460012","url":null,"abstract":"Given a graph [Formula: see text], a set of [Formula: see text] source-sink pairs [Formula: see text] [Formula: see text] and a profit bound [Formula: see text], every edge [Formula: see text] has a cost [Formula: see text], and every source-sink pair [Formula: see text] has a profit [Formula: see text] and a penalty [Formula: see text]. The [Formula: see text]-prize-collecting multicut problem ([Formula: see text]-PCMP) is to find a multicut [Formula: see text] such that the objective cost, which consists of the total cost of the edges in [Formula: see text] and the total penalty of the pairs still connected after removing [Formula: see text], is minimized and the total profit of the disconnected pairs by removing [Formula: see text] is at least [Formula: see text]. In this paper, we firstly consider the [Formula: see text]-PCMP in paths, and prove that it is [Formula: see text]-hard even when [Formula: see text] for any [Formula: see text]. Then, we present a fully polynomial time approximation scheme (FPTAS) whose running time is [Formula: see text] for the [Formula: see text]-PCMP in paths. Based on this algorithm, we present an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in spider graphs, and an FPTAS whose running time is [Formula: see text] for the [Formula: see text]-PCMP in rings, respectively, where [Formula: see text] is the number of leaves of spider graph.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45350162","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 : 2023-03-21DOI: 10.1142/s0129054123500077
Zeynep Nihan Berberler, A. Aytaç
Networks are known to be prone to node or link failures. A central issue in the analysis of networks is the assessment of their stability and reliability. A central concept that is used to assess stability and robustness of the performance of a network under failures is that of vulnerability. Node and link residual closeness are novel sensitive graph based characteristics for network vulnerability analysis. Node and link residual closeness measure the vulnerability even when the removal of nodes or links does not disconnect the network. Node and link residual closeness are of great theoretical and practical significance to network design and optimization. In this paper, vulnerabilities of multipartite network type topologies to the failure of individual nodes and links are computed via node and link residual closeness which provides a much fuller characterization of the network. Then, how multipartite network type topologies perform when they suffer a node or a link failure is analyzed.
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