Pub Date : 2026-01-17DOI: 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.
{"title":"Undecidability of the emptiness problem for weak models of distributed computing","authors":"Flavio T. Principato, Javier Esparza, Philipp Czerner","doi":"10.1016/j.tcs.2026.115757","DOIUrl":"10.1016/j.tcs.2026.115757","url":null,"abstract":"<div><div>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 <em>emptiness problem</em>: 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.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115757"},"PeriodicalIF":1.0,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189307","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 : 2026-01-17DOI: 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 X⊆V(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 -complete on graphs of diameter 4.
We design a polynomial-time algorithm that finds a μ-set of size , 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 , the mutual-visibility and dual mutual-visibility problems are not approximable within a factor of on graphs of diameter at least 3, while the outer and total mutual-visibility problems are not approximable within a factor of , 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。
{"title":"On the approximability of graph visibility problems","authors":"Davide Bilò, Alessia Di Fonso, Gabriele Di Stefano, Stefano Leucci","doi":"10.1016/j.tcs.2026.115766","DOIUrl":"10.1016/j.tcs.2026.115766","url":null,"abstract":"<div><div>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 <em>mutual-visibility</em> problem in a graph <em>G</em> of <em>n</em> vertices asks to find the largest set of vertices <em>X</em>⊆<em>V</em>(<em>G</em>), also called <em>μ</em>-set, such that for any two vertices <em>u, v</em> ∈ <em>X</em>, there is a shortest <em>u, v</em>-path <em>P</em> where all internal vertices of <em>P</em> are not in <em>X</em>. This means that <em>u</em> and <em>v</em> are visible w.r.t. <em>X</em>. Variations of this problem are known as <em>total, outer</em>, and <em>dual</em> mutual-visibility problems, depending on the visibility property of vertices inside and/or outside <em>X</em>. The mutual-visibility problem and all its variants are known to be <span><math><mi>NP</mi></math></span>-complete on graphs of diameter 4.</div><div>We design a polynomial-time algorithm that finds a <em>μ</em>-set of size <span><math><mrow><mstyle><mi>Ω</mi></mstyle><mrow><mo>(</mo><msqrt><mrow><mi>n</mi><mo>/</mo><mi>D</mi></mrow></msqrt><mo>)</mo></mrow></mrow></math></span>, where <em>D</em> is the average distance in <em>G</em>, 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 <span><math><mrow><mi>P</mi><mo>≠</mo><mi>NP</mi></mrow></math></span>, the mutual-visibility and dual mutual-visibility problems are not approximable within a factor of <span><math><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn><mo>−</mo><mrow><mi>ε</mi></mrow></mrow></msup></math></span> on graphs of diameter at least 3, while the outer and total mutual-visibility problems are not approximable within a factor of <span><math><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mrow><mi>ε</mi></mrow></mrow></msup></math></span>, 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 <em>u, v, w</em> of <em>X</em> belong to any shortest path of <em>G</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115766"},"PeriodicalIF":1.0,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039654","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 : 2026-01-17DOI: 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).
{"title":"Decision problems on geometric tilings","authors":"Benjamin Hellouin De Menibus , Victor H. Lutfalla , Pascal Vanier","doi":"10.1016/j.tcs.2026.115756","DOIUrl":"10.1016/j.tcs.2026.115756","url":null,"abstract":"<div><div>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).</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115756"},"PeriodicalIF":1.0,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039624","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 : 2026-01-15DOI: 10.1016/j.tcs.2026.115764
Tota Yamada, Yonghwan Kim, Yoshiaki Katayama
A Minus Dominating (MD) Function of a graph is a function that assigns a value from 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 , 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 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 rounds, where D is the diameter and Δ is the maximum degree of the graph. Each vertex requires 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。
{"title":"k-minimal minus domination and self-stabilization","authors":"Tota Yamada, Yonghwan Kim, Yoshiaki Katayama","doi":"10.1016/j.tcs.2026.115764","DOIUrl":"10.1016/j.tcs.2026.115764","url":null,"abstract":"<div><div>A <em>Minus Dominating (MD) Function</em> of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math></span><span><math><mrow><mo>(</mo><mo>|</mo><mi>V</mi><mo>|</mo><mo>=</mo><mi>n</mi><mo>)</mo></mrow></math></span> is a function that assigns a value from <span><math><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span> to each vertex <em>i</em> ∈ <em>V</em> such that the sum of the values of vertex <em>i</em> and all its neighboring vertices is positive (i.e., equal to or greater than 1). An MD function is <em>minimal</em> if decreasing the value of any vertex by 1 violates the conditions of the MD function.</div><div>As an extension of the MD function, we introduce the <em>k</em>-<em>Minimal Minus Dominating (MMD) Function</em> <span><math><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>, 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 <em>k</em> while decreasing the values of some vertices by at least <span><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> in total. According to the definition, any minimal MD function corresponds to a 0-MMD function.</div><div>In this paper, we propose a silent self-stabilizing algorithm to solve the <em>1-Minimal Minus Domination Problem</em>on an arbitrary graph. This algorithm employs a composition technique, known as <em>loop composition</em>, which repeatedly applies several self-stabilizing algorithms in order. The algorithm converges within <span><math><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mrow><mo>(</mo><msup><mstyle><mi>Δ</mi></mstyle><mn>2</mn></msup><mo>+</mo><mi>D</mi><mo>)</mo></mrow><mo>)</mo></mrow></math></span> rounds, where <em>D</em> is the diameter and Δ is the maximum degree of the graph. Each vertex requires <span><math><mrow><mi>O</mi><mo>(</mo><msup><mstyle><mi>Δ</mi></mstyle><mn>4</mn></msup><mi>log</mi><mi>n</mi><mo>)</mo></mrow></math></span> bits of memory.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115764"},"PeriodicalIF":1.0,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039650","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 : 2026-01-15DOI: 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.
{"title":"Graph spectrum of neighbourhood sombor matrix and structure-Property modelling","authors":"Sourav Mondal , Parikshit Das , Zahid Raza , Anita Pal , Modjtaba Ghorbani","doi":"10.1016/j.tcs.2026.115758","DOIUrl":"10.1016/j.tcs.2026.115758","url":null,"abstract":"<div><div>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 (<em>ζ</em><sub>1</sub>) 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 <em>ζ</em><sub>1</sub> in structure-property modelling is also demonstrated. The isomer-discrimination ability of <em>ζ</em><sub>1</sub> is found to be better than that of some well-known descriptors.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115758"},"PeriodicalIF":1.0,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081151","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}
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 -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 -hardness by showing a matching lower bound to our algorithmic result.
{"title":"Generalized capacity planning for the hospital-Residents problem","authors":"Haricharan Balasundaram , Girija Limaye , Meghana Nasre , Abhinav Raja","doi":"10.1016/j.tcs.2026.115760","DOIUrl":"10.1016/j.tcs.2026.115760","url":null,"abstract":"<div><div>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 <em>all</em> agents must be matched, we propose and study a generalized capacity planning problem, which allows cost-controlled flexibility with respect to quotas.</div><div>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.</div><div>We show that there is a sharp contrast – minimizing the local objective is polynomial-time solvable, whereas minimizing the global objective is <span><math><mi>NP</mi></math></span>-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 <span><math><mi>NP</mi></math></span>-hardness by showing a matching lower bound to our algorithmic result.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115760"},"PeriodicalIF":1.0,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080048","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 : 2026-01-14DOI: 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.
{"title":"On countable SFT covers of sparse multidimensional shift spaces","authors":"Ilkka Törmä","doi":"10.1016/j.tcs.2026.115755","DOIUrl":"10.1016/j.tcs.2026.115755","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115755"},"PeriodicalIF":1.0,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038909","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 : 2026-01-13DOI: 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 -time algorithm for solving it. Since the input size is , our algorithm is almost optimal.
{"title":"Computing the center of uncertain points on cactus graphs","authors":"Ran Hu , Divy H. Kanani , Jingru Zhang","doi":"10.1016/j.tcs.2026.115761","DOIUrl":"10.1016/j.tcs.2026.115761","url":null,"abstract":"<div><div>In this paper, we consider the (weighted) one-center problem of uncertain points on cactus graphs. Given are a cactus graph <em>G</em> and a set of <em>n</em> uncertain points. Each uncertain point has <em>m</em> possible locations on <em>G</em> with probabilities and a non-negative weight. The (weighted) one-center problem aims to compute a point (the center) <em>x</em>* on <em>G</em> to minimize the maximum (weighted) expected distance from <em>x</em>* to all uncertain points. No previous algorithms are known for this problem. In this paper, we propose an <span><math><mrow><mi>O</mi><mo>(</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>+</mo><mi>m</mi><mi>n</mi><mi>log</mi><mi>m</mi><mi>n</mi><mo>)</mo></mrow></math></span>-time algorithm for solving it. Since the input size is <span><math><mrow><mi>O</mi><mo>(</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>+</mo><mi>m</mi><mi>n</mi><mo>)</mo></mrow></math></span>, our algorithm is almost optimal.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115761"},"PeriodicalIF":1.0,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145993600","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 : 2026-01-12DOI: 10.1016/j.tcs.2026.115759
Thomas Dissaux, Nicolas Nisse
A path-decomposition of a graph 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 unless ). 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 . Our main result is a -approximation algorithm for the pathlength of outerplanar graphs. This algorithm is based on a characterization of almost optimal (of length at most ) path-decompositions of outerplanar graphs.
{"title":"Pathlength of outerplanar graphs","authors":"Thomas Dissaux, Nicolas Nisse","doi":"10.1016/j.tcs.2026.115759","DOIUrl":"10.1016/j.tcs.2026.115759","url":null,"abstract":"<div><div>A <em>path-decomposition</em> of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math></span> is a sequence of subsets of <em>V</em>, called <em>bags</em>, that satisfy some connectivity properties. The <em>length</em> of a path-decomposition of a graph <em>G</em> is the greatest distance (in <em>G</em>) between two vertices that belong to a same bag and the <em>pathlength</em>, denoted by <em>p</em>ℓ(<em>G</em>), of <em>G</em> 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, <em>etc</em>. However, deciding if the pathlength of a graph <em>G</em> is at most 2 is NP-complete, and the best known approximation algorithm has a ratio 2 (there is no <em>c</em>-approximation with <span><math><mrow><mi>c</mi><mo><</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span> unless <span><math><mrow><mi>P</mi><mo>=</mo><mi>N</mi><mi>P</mi></mrow></math></span>). 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 <em>n</em> vertices is equal to <span><math><mrow><mo>⌊</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo>⌋</mo></mrow></math></span>. Our main result is a <span><math><mrow><mo>(</mo><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-approximation algorithm for the pathlength of outerplanar graphs. This algorithm is based on a characterization of almost optimal (of length at most <span><math><mrow><mi>p</mi><mi>ℓ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mn>1</mn></mrow></math></span>) path-decompositions of outerplanar graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115759"},"PeriodicalIF":1.0,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980987","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}
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.
{"title":"Balanced substructures in bicolored graphs","authors":"P.S. Ardra , R. Krithika , Saket Saurabh , Roohani Sharma","doi":"10.1016/j.tcs.2026.115745","DOIUrl":"10.1016/j.tcs.2026.115745","url":null,"abstract":"<div><div>An edge-colored graph is said to be <em>balanced</em> if it has an equal number of edges of each color. Given a graph <em>G</em> whose edges are colored using two colors and a positive integer <em>k</em>, the objective in the <span>Edge Balanced Connected Subgraph</span> problem is to determine if <em>G</em> has a balanced connected subgraph containing at least <em>k</em> edges. We first show that this problem is <span>NP</span>-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 <span>Edge Balanced Connected Subgraph</span> and its variants (where the balanced subgraph is required to be a path/tree) with respect to <em>k</em> as the parameter. Towards this, we show that if a graph has a balanced connected subgraph/tree/path of size at least <em>k</em>, then it has one of size at least <em>k</em> and at most <em>f</em>(<em>k</em>) where <em>f</em> is a linear function. We use this result combined with dynamic programming algorithms based on <em>color coding</em> and <em>representative sets</em> to show that <span>Edge Balanced Connected Subgraph</span> and its variants are <span>FPT</span>. Further, using polynomial-time reductions to the <span>Multilinear Monomial Detection</span> problem, we give faster randomized <span>FPT</span> algorithms for the problems. In order to describe these reductions, we define a combinatorial object called <em>relaxed-subgraph</em>. 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 <span>Steiner Tree</span> problem and may be of independent interest.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115745"},"PeriodicalIF":1.0,"publicationDate":"2026-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980988","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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