Theoretical Computer Science最新文献

{"title":"Online multi-dimensional generalized assignment problem with predictions","authors":"Yimeng Xu ,&nbsp;Jiaqi Zheng ,&nbsp;Guihai Chen ,&nbsp;Xia Zhu ,&nbsp;Zhen Yao","doi":"10.1016/j.tcs.2025.115505","DOIUrl":"10.1016/j.tcs.2025.115505","url":null,"abstract":"<div><div>The Online Multi-Dimensional Generalized Assignment Problem (online MDGAP) can model a large number of applications such as parallel machine scheduling, vehicle routing, telecommunication network design, <em>etc.</em>, where a set of jobs have to be assigned to a set of capacitated agents in an online manner such that multi-dimensional capacity constraints can be respected. In this paper, we initiate the study of online MDGAP with predictions — the decision parameters such as coefficients of service (switching) costs and resource consumption can be accurately predicted or error-bounded, with the objective of minimizing the sum of service costs and switching costs in the long run. Furthermore, we design a two-stage online algorithm with performance guarantees. Rigorous theoretical analysis in terms of competitive ratio and regret demonstrates that our algorithm can produce an integer solution in polynomial time with bounded dimension constraints violation, robust to the coefficient variations and resource consumption uncertainty. Finally, trace-driven simulations show that our algorithm can achieve near optimal, high utilization, low constraint violation, and strong robustness.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1056 ","pages":"Article 115505"},"PeriodicalIF":1.0,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144813848","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}

{"title":"APTAS for bin packing with general cost structures","authors":"G. Jaykrishnan,&nbsp;Asaf Levin","doi":"10.1016/j.tcs.2025.115510","DOIUrl":"10.1016/j.tcs.2025.115510","url":null,"abstract":"<div><div>We consider the following generalization of the bin packing problem. We are given a set of items each of which is associated with a rational size in the interval <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, and a monotone non-decreasing non-negative cost function <em>f</em> defined over the cardinalities of the subsets of items. A feasible solution is a partition of the set of items into bins subject to the constraint that the total size of items in every bin is at most 1. Unlike bin packing, the goal function is to minimize the total cost of the bins where the cost of a bin is the value of <em>f</em> applied on the cardinality of the subset of items packed into the bin. We present an APTAS for this strongly NP-hard problem. We also provide a complete complexity classification of the problem with respect to the choice of <em>f</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115510"},"PeriodicalIF":1.0,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144809399","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}

{"title":"Edge-bipancyclicity of hypercubes with faulty edges","authors":"Qianhong Liu,&nbsp;Fan Wang","doi":"10.1016/j.tcs.2025.115503","DOIUrl":"10.1016/j.tcs.2025.115503","url":null,"abstract":"<div><div>In this paper, we consider the edge-bipancyclicity property of hypercubes with faulty edges. For <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, let <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be an <em>n</em>-dimensional hypercube, and let <em>F</em> be a set of faulty edges in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> such that <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≤</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>11</mn></math></span>. If every vertex in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>F</mi></math></span> has a degree of at least 2 and no 4-cycle in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>F</mi></math></span> has a pair of non-adjacent vertices both of degree 2, then every edge <em>e</em> in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>F</mi></math></span> lies on a fault-free cycle of every even length from 8 to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>, with two exceptions: (1) there exists a 4-cycle in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>F</mi></math></span> containing the edge <em>e</em> such that the vertices on the 4-cycle which are not incident with <em>e</em> both have a degree of 2; (2) there exist two vertices of degree 2 which are not endpoints of edge <em>e</em> but are adjacent to the same endpoint of <em>e</em> in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><mi>F</mi></math></span>. This result improves upon some known results on edge-bipancyclicity of hypercubes with faulty edges.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1056 ","pages":"Article 115503"},"PeriodicalIF":1.0,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144813847","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}

{"title":"Growth of bilinear maps III: decidability","authors":"Vuong Bui","doi":"10.1016/j.tcs.2025.115515","DOIUrl":"10.1016/j.tcs.2025.115515","url":null,"abstract":"<div><div>The following notion of growth rate can be seen as a generalization of joint spectral radius: Given a bilinear map <span><math><mo>⁎</mo><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with nonnegative coefficients and a nonnegative vector <span><math><mi>s</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, denote by <span><math><mi>g</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> the largest possible entry of a vector obtained by combining <em>n</em> instances of <em>s</em> using <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> applications of ⁎. Let <em>λ</em> denote the growth rate <span><math><msub><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>sup</mi></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mspace></mspace><mroot><mrow><mi>g</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mroot></math></span>. Rosenfeld showed that the problem of checking <span><math><mi>λ</mi><mo>≤</mo><mn>1</mn></math></span> is undecidable by reducing the problem of joint spectral radius.</div><div>In this article, we provide a simpler reduction using the observation that matrix multiplication is actually a bilinear map. Moreover, we extend the reduction to show that checking <span><math><mi>λ</mi><mo>≤</mo><mn>1</mn></math></span> is still undecidable even if <em>s</em> is positive. If there is no restriction on the signs, we can also show that the problem of checking if the system can produce a zero vector is undecidable by reducing the problem of checking the mortality of a pair of matrices. This answers a question asked by Rosenfeld. Beside that, we confirm a remark of Rosenfeld that the problem does not become harder when we introduce more bilinear maps and more starting vectors.</div><div>It is known that if the vector <em>s</em> is strictly positive, then the limit superior <em>λ</em> is actually a limit. However, we show that when <em>s</em> is only nonnegative, the problem of checking the existence of the limit is undecidable. This also answers a question asked by Rosenfeld.</div><div>We provide a formula for the growth rate <em>λ</em> in terms of the diagonals of matrices corresponding to a special structure called “linear pattern”. A condition is given so that the limit <em>λ</em> exists. This actually provides a simpler proof for the existence of the limit <em>λ</em> when <span><math><mi>s</mi><mo>&gt;</mo><mn>0</mn></math></span>. An important corollary of the formula is the computability of the growth rate, which answers another question by Rosenfeld. Another corollary is that the problem of checking <span><math><mi>λ</mi><mo>≤</mo><mn>1</mn></math></span> is reducible to the problem of joint spectral radius, via the halting problem, i.e., the two problems are Turing equivalent","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1056 ","pages":"Article 115515"},"PeriodicalIF":1.0,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010300","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}

{"title":"Mobile agents on chordal graphs: Maximum independent set and beyond","authors":"Tanvir Kaur,&nbsp;Kaustav Paul,&nbsp;Kaushik Mondal","doi":"10.1016/j.tcs.2025.115490","DOIUrl":"10.1016/j.tcs.2025.115490","url":null,"abstract":"<div><div>We consider the problem of finding Maximum Independent Set (MaxIS) for chordal graphs using mobile agents. Suppose <em>n</em> agents are initially placed arbitrarily on the nodes of an <em>n</em>-node chordal graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span>. Agents need to find a Maximum Independent Set <em>M</em> of <em>G</em> such that each node of <em>M</em> is occupied by at least one agent. Also, each of the <em>n</em> agents must know whether its occupied node is a part of <em>M</em> or not. We provide distributed algorithms for <em>n</em> mobile agents, each having <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> memory, to compute MaxIS of <em>G</em> in <span><math><mi>O</mi><mo>(</mo><mi>m</mi><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>Δ</mi><mo>)</mo></math></span> time, where <em>m</em> denotes the number of edges in <em>G</em>, <em>n</em> denotes the number of nodes in <em>G</em>, and Δ is the maximum degree of the graph. At first, we design an algorithm considering the case where all agents are initially placed at the same node (i.e., rooted initial configuration). To run this algorithm, the agents do not require prior knowledge of any global parameter. Then we propose an algorithm for the case where agents are initially distributed arbitrarily across the graph (i.e., arbitrary initial configuration). To run this algorithm, the agents require prior knowledge of certain global parameters. Further, we provide faster algorithms for finding MaxIS in chordal graphs either by increasing the memory available to each agent or by employing more agents. We report that by using a similar approach, it is possible to find the maximum clique in chordal graphs and color any chordal graph with the minimum number of colors. We also provide a dynamic programming-based distributed algorithm to find a Maximum Independent Set for trees in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> time.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115490"},"PeriodicalIF":1.0,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780771","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}

{"title":"Target influence maximization problem under avoiding unexpected users interruption in online social network","authors":"Xiaoping Zhu ,&nbsp;Jianming Zhu ,&nbsp;Priyanshi Garg ,&nbsp;Guoqing Wang","doi":"10.1016/j.tcs.2025.115513","DOIUrl":"10.1016/j.tcs.2025.115513","url":null,"abstract":"<div><div>The effectiveness of viral marketing lies in the fact that it can activate a “chain-reaction” driven by word-of-mouth. This paper defines unexpected users’ interruption as the unintentional activation of impedimental users who disseminate negative evaluations. Such interruptions may arise during the process of selecting seed nodes intended to maximize influence spread among target users and foster brand awareness. Furthermore, it is typical to tailor promotions for specific user groups. For instance, consider an event organizer who shares an event advertisement on a social platform, aiming to capture the interest of a large local audience. To solve this problem, we devise positive and target influence maximization under the impedimental influence problem (PTIM), aiming to initiate from a set of influential seed users such that eventually, they will trigger the largest positive influence spread among target users. We prove that the objective function is neither submodular nor supermodular. Then, we decompose it into the difference between two submodular functions (DS decomposition). To solve the maximization problem of DS decomposition under cardinality constraint, we provide an explicit DS greedy algorithm with approximation guarantee. Finally, the experimental simulations on three network data sets demonstrate the effectiveness of our method.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1056 ","pages":"Article 115513"},"PeriodicalIF":1.0,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144852209","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}

{"title":"Visualizing differentially private mechanisms with physical cards","authors":"Reo Eriguchi ,&nbsp;Kazumasa Shinagawa ,&nbsp;Takao Murakami","doi":"10.1016/j.tcs.2025.115492","DOIUrl":"10.1016/j.tcs.2025.115492","url":null,"abstract":"<div><div>Card-based cryptography studies the problem of implementing cryptographic algorithms in a visual way using physical cards to demonstrate their security properties for those who are unfamiliar with cryptography. In this paper, we initiate the study of card-based implementations of differentially private mechanisms, which are a standard privacy-enhancing technique to publish statistics of databases. We start with giving the definition of differential privacy of card-based protocols. As a feasibility result, we present three kinds of protocols using standard binary cards for computing the sum of parties' binary inputs, <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <figure><img></figure>, under differential privacy. Our first protocol follows the framework of output perturbation, which provides differential privacy by adding noise to exact aggregation results. The protocol needs only two shuffles, and the overheads in the number of cards and the error bound are independent of the number <em>n</em> of parties. Our second and third protocols are based on Randomized Response, which adds noise to each input before aggregation. Compared to the first protocol, they improve the overheads in the number of cards and the error bound in terms of differential privacy parameters. Furthermore, we show that our card-based noise generation method can be used to compute general functions beyond the binary sum under differential privacy. To address a technical challenge of generating non-uniform noise using a finite number of cards, we propose a novel differentially private mechanism based on the hypergeometric distribution, which we believe may be of independent interest beyond applications to card-based cryptography.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115492"},"PeriodicalIF":1.0,"publicationDate":"2025-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144750444","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}

{"title":"Deterministic protocols for Voronoi diagrams and triangulations of planar point sets on the congested clique","authors":"Jesper Jansson ,&nbsp;Christos Levcopoulos ,&nbsp;Andrzej Lingas ,&nbsp;Valentin Polishchuk ,&nbsp;Quan Xue","doi":"10.1016/j.tcs.2025.115491","DOIUrl":"10.1016/j.tcs.2025.115491","url":null,"abstract":"<div><div>We study the problems of computing the Voronoi diagram and a triangulation of a set of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> points with <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-bit coordinates in the Euclidean plane in a substantially sublinear in <em>n</em> number of rounds in the congested clique model with <em>n</em> nodes. First, we observe that if the points are uniformly at random distributed in a unit square then their Voronoi diagram within the square can be computed in <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> rounds with high probability (w.h.p.). Next, we show that if a very weak smoothness condition is satisfied by an input set of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> points with <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-bit coordinates in the unit square then the Voronoi diagram of the point set within the unit square can be deterministically computed in <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> rounds in this model. Finally, we present a deterministic <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-round protocol for a triangulation of <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> points with <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-bit coordinates in the Euclidean plane. It relies on our novel method for extending triangulations of two planar point sets separated by a straight line to a complete triangulation of the union of the sets in <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> rounds.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115491"},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738333","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}

{"title":"Competitive mechanisms for energy-efficient cloud computing","authors":"Antonios Antoniadis ,&nbsp;Andrés Cristi ,&nbsp;Tim Oosterwijk ,&nbsp;Alkmini Sgouritsa","doi":"10.1016/j.tcs.2025.115488","DOIUrl":"10.1016/j.tcs.2025.115488","url":null,"abstract":"<div><div>We present a general model for the operation of a cloud computing server comprised of one or more speed-scalable processors. Typically, agents submit tasks to such a cloud computing server in an online fashion, and the server operator has to schedule the tasks and decide on payments without knowledge of tasks arriving in the future. Moreover, the operator should take the different incentives of the agents into account and aim to minimize the energy expenditure. For both the offline and the online setting we provide mechanisms with several desirable properties: The induced game admits a Nash equilibrium, the mechanism is budget balanced, has low communication complexity, is computationally tractable, is intuitive to explain, but above all, has a constant Price of Anarchy. Therefore, the total costs are not too far off from the social optimum. We extend our results to the case of multiple processors and to the Bayesian setting.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115488"},"PeriodicalIF":1.0,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144766569","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}

{"title":"Packing internally disjoint Steiner paths of modified bubble-sort networks","authors":"Lina Zhao ,&nbsp;Shiying Wang ,&nbsp;Wei Feng","doi":"10.1016/j.tcs.2025.115484","DOIUrl":"10.1016/j.tcs.2025.115484","url":null,"abstract":"<div><div>Let <em>G</em> be a connected simple graph with vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and edge set <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. For <span><math><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mn>2</mn></math></span>, a path <em>P</em> in <em>G</em> is said to be an <em>S</em>-Steiner path (or <em>S</em>-path for short) if it connects all vertices of <em>S</em>. Two <em>S</em>-paths <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> are internally disjoint if <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>∩</mo><mi>E</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> and <span><math><mi>V</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>∩</mo><mi>V</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>S</mi></math></span>. The packing number of internally disjoint <em>S</em>-paths, denoted as <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo></math></span>, is the maximum number of internally disjoint <em>S</em>-paths in <em>G</em>. For an integer <span><math><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math></span>, the <em>k</em>-path connectivity of a graph <em>G</em> is defined as <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo></math></span> min<span><math><mo>{</mo><msub><mrow><mi>π</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>=</mo><mi>k</mi><mo>}</mo></math></span>. The modified bubble-sort graph, denoted <span><math><mi>M</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, is an interconnection network topological model for multiprocessor systems. In this paper, we focus on the 3-path-connectivity of the <em>n</em>-dimensional modified bubble sort graphs. By analyzing the structural properties of <span><math><mi>M</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, we determine that <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>M</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>⌊</mo><mfrac><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1055 ","pages":"Article 115484"},"PeriodicalIF":1.0,"publicationDate":"2025-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738334","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}