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Undecidability of the emptiness problem for weak models of distributed computing 分布式计算弱模型空性问题的不可判定性
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-17 DOI: 10.1016/j.tcs.2026.115757
Flavio T. Principato, Javier Esparza, Philipp Czerner
Esparza and Reiter have recently conducted a systematic comparative study of weak asynchronous models of distributed computing, in which a network of identical finite-state machines acts cooperatively to decide properties of the network’s graph. They introduced a distributed automata framework encompassing many different models, and proved that w.r.t. their expressive power (the graph properties they can decide) distributed automata collapse into seven equivalence classes. In this contribution, we turn our attention to the formal verification problem: Given a distributed automaton, does it decide a given graph property? We consider a fundamental instance of this question – the emptiness problem: Given a distributed automaton, does it accept any graph at all? Our main result is negative: the emptiness problem is undecidable for six of the seven equivalence classes, and trivially decidable for the remaining class.
Esparza和Reiter最近对分布式计算的弱异步模型进行了系统的比较研究,在这种模型中,相同的有限状态机组成的网络协作决定网络图的属性。他们引入了一个包含许多不同模型的分布式自动机框架,并证明了它们的表达能力(它们可以决定的图属性),分布式自动机可以分解成七个等价类。在本文中,我们将注意力转向形式化验证问题:给定一个分布式自动机,它是否决定给定的图属性?我们考虑这个问题的一个基本实例——空性问题:给定一个分布式自动机,它是否接受任何图?我们的主要结果是否定的:对于7个等价类中的6个来说,空性问题是不可判定的,对于剩下的类来说,空性问题是可判定的。
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引用次数: 0
On the approximability of graph visibility problems 图可见性问题的近似性
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-17 DOI: 10.1016/j.tcs.2026.115766
Davide Bilò, Alessia Di Fonso, Gabriele Di Stefano, Stefano Leucci
Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph G of n vertices asks to find the largest set of vertices XV(G), also called μ-set, such that for any two vertices u, v ∈ X, there is a shortest u, v-path P where all internal vertices of P are not in X. This means that u and v are visible w.r.t. X. Variations of this problem are known as total, outer, and dual mutual-visibility problems, depending on the visibility property of vertices inside and/or outside X. The mutual-visibility problem and all its variants are known to be NP-complete on graphs of diameter 4.
We design a polynomial-time algorithm that finds a μ-set of size Ω(n/D), where D is the average distance in G, we show inapproximability results for all visibility problems on graphs of diameter 2, and we strengthen the inapproximability ratios for graphs of diameter 3 or larger. More precisely, assuming PNP, the mutual-visibility and dual mutual-visibility problems are not approximable within a factor of n1/3ε on graphs of diameter at least 3, while the outer and total mutual-visibility problems are not approximable within a factor of n1/2ε, for any constant ε > 0. Finally, we study the relationship between the mutual-visibility number and the general position number, in which no three distinct vertices u, v, w of X belong to any shortest path of G.
由于可见性问题是具有挑战性的组合问题,并且与机器人导航问题密切相关,因此长期以来人们在不同的假设下对其进行了研究。有n个顶点的图G中的互可见性问题要求求出最大的顶点集X V(G),也称为μ-set,使得对于任意两个顶点u, V ∈ X,存在一个最短的u, V路径P,且P的所有内部顶点都不在X中。这意味着u和V在w.r.t.x中是可见的。这个问题的变体被称为总、外和对偶互可见性问题。依赖于x内外顶点的可见性,互可见性问题及其所有变体在直径为4的图上已知是np完全的。我们设计了一个多项式时间算法,它找到一个大小为Ω(n/D)的μ集,其中D是G中的平均距离,我们在直径为2的图上展示了所有可见性问题的不近似结果,并且我们加强了直径为3或更大的图的不近似比率。更准确地说,假设P≠NP,在直径至少为3的图上,互可视性和对偶互可视性问题在n1/3−ε因子范围内是不可近似的,而对于任意常数ε >; 0,外互可视性和全互可视性问题在n1/2−ε因子范围内是不可近似的。最后,我们研究了互可见数与一般位置数的关系,其中X的三个不同的顶点u, v, w不属于G的任何最短路径。
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引用次数: 0
Decision problems on geometric tilings 几何平铺的决策问题
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-17 DOI: 10.1016/j.tcs.2026.115756
Benjamin Hellouin De Menibus , Victor H. Lutfalla , Pascal Vanier
We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes, extending the results of [1] on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set. Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that this problem is undecidable even in a simple setting (square shapes with small modifications).
研究几何平铺的决策问题。首先,我们研究了多米诺问题的一个变体,其中正方形的瓷砖被任意形状的几何瓷砖所取代。我们证明,无论形状如何,这种变体都是不可确定的,将[1]的结果扩展到菱形瓷砖上。即使几何平铺被强制属于一个固定的集合,这个结果也成立。其次,我们考虑了几何子移是否具有有限局部复杂度的问题,这是研究几何平铺时的一个常见假设。我们表明,即使在一个简单的设置(小修改的方形)中,这个问题也是不可确定的。
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引用次数: 0
k-minimal minus domination and self-stabilization 最小负控制与自稳定
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-15 DOI: 10.1016/j.tcs.2026.115764
Tota Yamada, Yonghwan Kim, Yoshiaki Katayama
A Minus Dominating (MD) Function of a graph G=(V,E)(|V|=n) is a function that assigns a value from {1,0,1} to each vertex i ∈ V such that the sum of the values of vertex i and all its neighboring vertices is positive (i.e., equal to or greater than 1). An MD function is minimal if decreasing the value of any vertex by 1 violates the conditions of the MD function.
As an extension of the MD function, we introduce the k-Minimal Minus Dominating (MMD) Function (0k2n1), which is a minimal MD function with additional condition such that no other MD function can be obtained by increasing the values of some vertices by a total of k while decreasing the values of some vertices by at least k+1 in total. According to the definition, any minimal MD function corresponds to a 0-MMD function.
In this paper, we propose a silent self-stabilizing algorithm to solve the 1-Minimal Minus Domination Problemon an arbitrary graph. This algorithm employs a composition technique, known as loop composition, which repeatedly applies several self-stabilizing algorithms in order. The algorithm converges within O(n(Δ2+D)) rounds, where D is the diameter and Δ is the maximum degree of the graph. Each vertex requires O(Δ4logn) bits of memory.
图G=(V,E)(|V|=n)的负支配(MD)函数是这样一个函数,它给每个顶点i ∈ V赋一个{−1,0,1}的值,使得顶点i与其所有相邻顶点的值之和为正(即等于或大于1)。如果将任意顶点的值减少1违反了MD函数的条件,则MD函数是最小的。作为MD函数的扩展,我们引入了k- minimal - Minus (MMD) function(0≤k≤2n−1),这是一个最小MD函数,它带有附加条件,使得不能通过将某些顶点的值增加k而将某些顶点的值减少至少k+1来获得其他MD函数。根据定义,任何最小MD函数都对应一个0-MMD函数。本文提出了一种无声自稳定算法来解决任意图的1-极小负控制问题。该算法采用了一种复合技术,称为循环复合,它按顺序重复应用几个自稳定算法。算法在O(n(Δ2+D))轮内收敛,其中D为图的直径,Δ为图的最大度。每个顶点需要O(Δ4logn)位内存。
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引用次数: 0
Graph spectrum of neighbourhood sombor matrix and structure-Property modelling 邻域sombor矩阵图谱与结构-性质建模
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-15 DOI: 10.1016/j.tcs.2026.115758
Sourav Mondal , Parikshit Das , Zahid Raza , Anita Pal , Modjtaba Ghorbani
The spectral graph theory investigates the relationships between combinatorial qualities of graphs and algebraic properties of related matrices. The adjacency matrix is currently undergoing significant modification as a result of its well-developed theoretical and application standpoint. The present work deals with one such extension of the adjacency matrix. We propose here the neighborhood Sombor matrix corresponding to the well-known Sombor index. We compute the neighborhood Sombor spectrum of some benchmark graphs. Lower and upper bounds of the spectral radius (ζ1) are derived with identifying extremal graphs. Moreover, extremal trees are characterized in view of spectral radius, where path and star graphs yield minimal and maximal structures, respectively. The role of ζ1 in structure-property modelling is also demonstrated. The isomer-discrimination ability of ζ1 is found to be better than that of some well-known descriptors.
谱图理论研究图的组合性质与相关矩阵的代数性质之间的关系。邻接矩阵由于其成熟的理论和应用立场,目前正在经历重大的修改。本文研究邻接矩阵的一种扩展。我们在这里提出了邻域Sombor矩阵,对应于众所周知的Sombor指数。我们计算了一些基准图的邻域Sombor谱。用识别极值图导出了谱半径(ζ1)的下界和上界。此外,根据光谱半径对极值树进行表征,其中路径图和星图分别产生最小和最大结构。并论证了ζ1在构造-性质建模中的作用。ζ1的同分异构体识别能力优于一些已知的描述子。
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引用次数: 0
Generalized capacity planning for the hospital-Residents problem 医院-居民问题的广义容量规划
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-14 DOI: 10.1016/j.tcs.2026.115760
Haricharan Balasundaram , Girija Limaye , Meghana Nasre , Abhinav Raja
The Hospital Residents setting models important problems like school choice, assignment of undergraduate students to degree programs, among many others. In this setting, fixed quotas are associated with the programs that limit the number of agents that can be assigned to them. Motivated by scenarios where all agents must be matched, we propose and study a generalized capacity planning problem, which allows cost-controlled flexibility with respect to quotas.
Our setting is an extension of the Hospital Resident setting where programs have the usual quota as well as an associated cost, indicating the cost of matching an agent beyond the initial quotas. We seek to compute a matching that matches all agents and is optimal with respect to preferences, and minimizes either a local or a global objective on cost.
We show that there is a sharp contrast – minimizing the local objective is polynomial-time solvable, whereas minimizing the global objective is NP-hard. On the positive side, we present approximation algorithms for the global objective in the general case and a particular hard case. We achieve the approximation guarantee for the special hard case via a linear programming based algorithm. We strengthen the NP-hardness by showing a matching lower bound to our algorithmic result.
《住院医师》为许多重要问题树立了榜样,比如学校选择、本科生学位课程的分配等等。在此设置中,固定配额与限制可以分配给它们的代理数量的程序相关联。在所有代理必须匹配的情况下,我们提出并研究了一个广义容量规划问题,该问题允许相对于配额的成本控制灵活性。我们的设置是医院住院医师设置的扩展,其中程序具有通常的配额和相关的成本,表明在初始配额之外匹配代理的成本。我们试图计算匹配所有代理的匹配,并且在偏好方面是最优的,并且最小化局部或全局目标的成本。我们证明了一个鲜明的对比——最小化局部目标是多项式时间可解的,而最小化全局目标是np困难的。在积极的方面,我们提出了在一般情况下和特殊困难情况下的全局目标的近似算法。通过一种基于线性规划的算法,实现了特殊情况下的近似保证。我们通过显示匹配的下界来增强我们的算法结果的np -硬度。
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引用次数: 0
On countable SFT covers of sparse multidimensional shift spaces 稀疏多维位移空间的可数SFT覆盖
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-14 DOI: 10.1016/j.tcs.2026.115755
Ilkka Törmä
A multidimensional sofic shift is called countably covered if it has an SFT cover containing only countably many configurations. In contrast to the one-dimensional setting, not all countable sofic shifts are countably covered. We investigate the existence of countable covers for gap width shifts, where the number of nonzero symbols in a configuration is bounded by a function of the minimum distance between two such symbols. As our main results, we characterize those one-dimensional gap width shifts whose two-dimensional lift is a countably covered sofic shift, and show that a large class of two-dimensional gap width shifts are countably covered.
如果一个多维softshift的SFT覆盖只包含可数个构型,那么它被称为可数覆盖。与一维设置相反,并非所有可数的软移都被可数覆盖。我们研究了间隙宽度位移的可数覆盖的存在性,其中构型中非零符号的数量由两个此类符号之间的最小距离的函数限定。我们的主要研究结果,刻画了二维升力为可数覆盖位移的一维隙宽位移,并证明了一类二维隙宽位移是可数覆盖的。
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引用次数: 0
Computing the center of uncertain points on cactus graphs 计算仙人掌图上不确定点的中心
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-13 DOI: 10.1016/j.tcs.2026.115761
Ran Hu , Divy H. Kanani , Jingru Zhang
In this paper, we consider the (weighted) one-center problem of uncertain points on cactus graphs. Given are a cactus graph G and a set of n uncertain points. Each uncertain point has m possible locations on G with probabilities and a non-negative weight. The (weighted) one-center problem aims to compute a point (the center) x* on G to minimize the maximum (weighted) expected distance from x* to all uncertain points. No previous algorithms are known for this problem. In this paper, we propose an O(|G|+mnlogmn)-time algorithm for solving it. Since the input size is O(|G|+mn), our algorithm is almost optimal.
本文研究仙人掌图上不确定点的(加权)单中心问题。给定仙人掌图G和n个不确定点的集合。每个不确定点在G上有m个可能的位置,具有概率和非负权。(加权)单中心问题旨在计算G上的一个点(中心)x*,以最小化从x*到所有不确定点的最大(加权)期望距离。以前没有已知的算法可以解决这个问题。在本文中,我们提出了一个O(|G|+mnlogmn)时间算法来求解它。由于输入大小为0 (|G|+mn),我们的算法几乎是最优的。
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引用次数: 0
Pathlength of outerplanar graphs 外平面图的路径长度
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-12 DOI: 10.1016/j.tcs.2026.115759
Thomas Dissaux, Nicolas Nisse
A path-decomposition of a graph G=(V,E) is a sequence of subsets of V, called bags, that satisfy some connectivity properties. The length of a path-decomposition of a graph G is the greatest distance (in G) between two vertices that belong to a same bag and the pathlength, denoted by pℓ(G), of G is the smallest length of its path-decompositions. This parameter has been studied for its algorithmic applications for several classical metric problems like the minimum eccentricity shortest path problem, the line-distortion problem, etc. However, deciding if the pathlength of a graph G is at most 2 is NP-complete, and the best known approximation algorithm has a ratio 2 (there is no c-approximation with c<32 unless P=NP). In this work, we focus on the study of the pathlength of simple sub-classes of planar graphs. We start by designing a linear-time algorithm that computes the pathlength of trees. Then, we show that the pathlength of cycles with n vertices is equal to n2. Our main result is a (+1)-approximation algorithm for the pathlength of outerplanar graphs. This algorithm is based on a characterization of almost optimal (of length at most p(G)+1) path-decompositions of outerplanar graphs.
图G=(V,E)的路径分解是V的子集序列,称为袋,满足某些连通性。图G的路径分解的长度是属于同一袋的两个顶点之间的最大距离(在G中),而G的路径长度表示为p (G),是其路径分解的最小长度。研究了该参数在若干经典度量问题中的算法应用,如最小偏心最短路径问题、线畸变问题等。然而,判断图G的路径长度是否最多为2是NP完全的,而最著名的近似算法的比值为2(除非P=NP,否则不存在c<;32的c近似)。在这项工作中,我们重点研究了平面图的简单子类的路径长度。我们首先设计一个线性时间算法来计算树的路径长度。然后,我们证明了有n个顶点的环的路径长度等于⌊n2⌋。我们的主要结果是外平面图路径长度的(+1)逼近算法。该算法基于外平面图的几乎最优(长度最多为p (G)+1)路径分解的表征。
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引用次数: 0
Balanced substructures in bicolored graphs 双色图中的平衡子结构
IF 1 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Pub Date : 2026-01-11 DOI: 10.1016/j.tcs.2026.115745
P.S. Ardra , R. Krithika , Saket Saurabh , Roohani Sharma
An edge-colored graph is said to be balanced if it has an equal number of edges of each color. Given a graph G whose edges are colored using two colors and a positive integer k, the objective in the Edge Balanced Connected Subgraph problem is to determine if G has a balanced connected subgraph containing at least k edges. We first show that this problem is NP-complete and remains so even if the solution is required to be a tree or a path. Then, we focus on the parameterized complexity of Edge Balanced Connected Subgraph and its variants (where the balanced subgraph is required to be a path/tree) with respect to k as the parameter. Towards this, we show that if a graph has a balanced connected subgraph/tree/path of size at least k, then it has one of size at least k and at most f(k) where f is a linear function. We use this result combined with dynamic programming algorithms based on color coding and representative sets to show that Edge Balanced Connected Subgraph and its variants are FPT. Further, using polynomial-time reductions to the Multilinear Monomial Detection problem, we give faster randomized FPT algorithms for the problems. In order to describe these reductions, we define a combinatorial object called relaxed-subgraph. We define this object in such a way that balanced connected subgraphs, trees and paths are relaxed-subgraphs with certain properties. This object is defined in the spirit of branching walks known for the Steiner Tree problem and may be of independent interest.
如果一个有不同颜色边的图有相同数量的边,我们就说它是平衡的。给定一个图G,它的边是用两种颜色和一个正整数k着色的,边缘平衡连通子图问题的目标是确定G是否有一个包含至少k条边的平衡连通子图。我们首先证明了这个问题是np完全的,并且即使要求解是树或路径也仍然是np完全的。然后,我们重点研究了以k为参数的边平衡连通子图及其变体(其中平衡子图要求为路径/树)的参数化复杂度。为此,我们证明,如果一个图有一个大小至少为k的平衡连通子图/树/路径,那么它就有一个大小至少为k,最大为f(k)的子图/树/路径,其中f是一个线性函数。我们将这一结果与基于颜色编码和代表集的动态规划算法相结合,证明边缘平衡连通子图及其变体是FPT的。此外,对多线性单项检测问题使用多项式时间约简,我们给出了更快的随机化FPT算法。为了描述这些约简,我们定义了一个称为松弛子图的组合对象。我们这样定义这个对象:平衡连接子图、树和路径是具有某些属性的松弛子图。这个对象是根据斯坦纳树问题中分支行走的精神来定义的,并且可能具有独立的兴趣。
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引用次数: 0
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Theoretical Computer Science
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