Theoretical Computer Science最新文献

{"title":"On the optimal mixing problem of approximate Nash equilibria in bimatrix games","authors":"Xiaotie Deng ,&nbsp;Dongchen Li ,&nbsp;Hanyu Li","doi":"10.1016/j.tcs.2025.115072","DOIUrl":"10.1016/j.tcs.2025.115072","url":null,"abstract":"<div><div>This paper introduces the optimal mixing problem, a natural extension of the computation of approximate Nash Equilibria (NE) in bimatrix games. The problem focuses on determining the optimal convex combination of given strategies that minimizes the approximation (i.e., regret) in NE computation. We develop algorithms for the exact and approximate optimal mixing problems and present new complexity results that bridge both practical and theoretical aspects of NE computation. Practically, our algorithms can be used to enhance and integrate arbitrary existing constant-approximate NE algorithms, offering a powerful tool for the design of approximate NE algorithms. Theoretically, these algorithms allow us to explore the implications of support restrictions on approximate NE and derive the upper-bound separations between approximate NE and exact NE. Consequently, this work contributes to theoretical understandings of the computational complexity of approximate NE under various constraints and practical improvements in multi-agent reinforcement learning (MARL) and other fields where NE computation is involved.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115072"},"PeriodicalIF":0.9,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265810","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":"Maximum independent set formation on a finite grid by myopic robots","authors":"Raja Das,&nbsp;Avisek Sharma,&nbsp;Buddhadeb Sau","doi":"10.1016/j.tcs.2025.115077","DOIUrl":"10.1016/j.tcs.2025.115077","url":null,"abstract":"<div><div>This work deals with the Maximum Independent Set (<span><math><mi>MAXIS</mi></math></span>) formation problem in a finite rectangular grid by autonomous robots. Suppose we are given a set of identical robots, where each robot is placed on a node of a finite rectangular grid <span><math><mi>G</mi></math></span> such that no two robots are on the same node. The <span><math><mi>MAXIS</mi></math></span> formation problem asks to design an algorithm and each robot will move autonomously after executing the algorithm and terminate at a node such that after a finite time the set of nodes occupied by the robots is a maximum independent set of <span><math><mi>G</mi></math></span>. We assume that robots are anonymous and they execute the same distributed algorithm.</div><div>Previous works solved this problem using one or several door nodes through which the robots enter the grid or the graph one by one and occupy the required nodes. In this work, we propose a deterministic algorithm that solves the <span><math><mi>MAXIS</mi></math></span> formation problem in a more generalized scenario, i.e., when the total number of required robots to form a <span><math><mi>MAXIS</mi></math></span> are arbitrarily placed on the grid. The proposed algorithm works under a semi-synchronous scheduler using robots with only two hop visibility range and only three colors.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115077"},"PeriodicalIF":0.9,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265844","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":"Private least absolute deviations with heavy-tailed data","authors":"Di Wang ,&nbsp;Jinhui Xu","doi":"10.1016/j.tcs.2025.115071","DOIUrl":"10.1016/j.tcs.2025.115071","url":null,"abstract":"<div><div>We study the problem of Differentially Private Stochastic Convex Optimization (DPSCO) with heavy-tailed data. Specifically, we focus on the problem of Least Absolute Deviations, i.e., <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-norm linear regression, in the <em>ϵ</em>-DP model. While most previous work focuses on the case where the loss function is Lipschitz, in this paper we only need to assume the variates have bounded moments. Firstly, we study the case where the <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> norm of data has a bounded second-order moment. We propose an algorithm that is based on the exponential mechanism and show that it is possible to achieve an upper bound of <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msqrt><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>ϵ</mi></mrow></mfrac></mrow></msqrt><mo>)</mo></math></span> (with high probability). Next, we relax the assumption to bounded <em>θ</em>-th order moment with some <span><math><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> and show that it is possible to achieve an upper bound of <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mi>ϵ</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mfrac><mrow><mi>θ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>θ</mi></mrow></mfrac></mrow></msup><mo>)</mo></math></span>. Our algorithms can also be extended to more relaxed cases where only each coordinate of the data has bounded moments, and we can get an upper bound of <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msqrt><mrow><mi>n</mi><mi>ϵ</mi></mrow></msqrt></mrow></mfrac><mo>)</mo></math></span> and <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mfrac><mrow><mi>d</mi></mrow><mrow><msup><mrow><mo>(</mo><mi>n</mi><mi>ϵ</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mi>θ</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>θ</mi></mrow></mfrac></mrow></msup></mrow></mfrac><mo>)</mo></math></span> in the second and <em>θ</em>-th moment case respectively.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115071"},"PeriodicalIF":0.9,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143305539","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":"On the computational and descriptional complexity of multi-pattern languages","authors":"Jingnan Xie ,&nbsp;Harry B. Hunt III ,&nbsp;Richard E. Stearns","doi":"10.1016/j.tcs.2025.115063","DOIUrl":"10.1016/j.tcs.2025.115063","url":null,"abstract":"<div><div>We study the complexity of a restricted version of the universality problem: testing equivalence to <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> for languages that are either <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> or <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mo>⁎</mo></mrow></msup><mo>−</mo><mo>{</mo><mi>w</mi><mo>}</mo></math></span> where <span><math><mi>w</mi><mo>∈</mo><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. We show that this restricted problem for multi-pattern languages (MPL) is NP-hard, and for multi-pattern languages with regular substitutions (MPL_REG) is not recursively enumerable. Moreover, this undecidability result holds for any class of languages that effectively contains <span><math><mo>{</mo><mi>w</mi><mi>#</mi><mi>w</mi><mspace></mspace><mo>|</mo><mspace></mspace><mi>w</mi><mo>∈</mo><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mo>⁎</mo></mrow></msup><mo>}</mo></math></span>, and is effectively closed under union, and concatenation with regular sets. Consequently, it has broad applicability and extends to many generalizations of regular languages. Sufficient conditions are then presented for a language predicate to be as hard as the restricted universality problem. By applying these conditions, we develop a uniform method for showing undecidability and complexity results simultaneously via highly efficient many-one reductions. In addition, the properties of this restricted universality problem can be used to investigate the descriptional complexity of multi-patterns. We show that the trade-off between multi-patterns and regular expressions is not exponential-size bounded. Non-recursive trade-offs between multi-patterns (or multi-patterns with regular substitutions) and numerous classes of language descriptors are also established.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115063"},"PeriodicalIF":0.9,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143306088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Evaluating the reliability of complete Josephus cubes under extra link fault with the optimal solution of the edge isoperimetric problem","authors":"Sufang Liu ,&nbsp;Zhaoman Huang ,&nbsp;Yueke Lv ,&nbsp;Chia-Wei Lee","doi":"10.1016/j.tcs.2025.115064","DOIUrl":"10.1016/j.tcs.2025.115064","url":null,"abstract":"<div><div>The <em>h</em>-extra edge-connectivity of a connected graph <em>G</em> <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, as a generalization of the classic Menger's Theorem, is the minimum number of edges that need to be removed to disconnect graph <em>G</em> and ensure that each component of the remaining graph has at least <em>h</em> vertices. This paper focuses on the reliability of the <em>h</em>-extra edge-connectivity of the complete Josephus cube <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> interconnection network, a variant of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, as the underlying topology, denoted as <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>. By analyzing the properties of the optimal solution of the edge isoperimetric problem on <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the exact value and the <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>-optimality of the <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo><mo>+</mo><mn>1</mn></mrow></msup></math></span> is characterized. For a sufficiently large positive integer <em>n</em>, about 44.44% of positive integers <em>h</em> in the well-defined interval <span><math><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, the corresponding <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> occurs a concentration phenomenon.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115064"},"PeriodicalIF":0.9,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143306089","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":"Feasibility analysis of recurrent DAG tasks is PSPACE-hard","authors":"Vincenzo Bonifaci ,&nbsp;Alberto Marchetti-Spaccamela","doi":"10.1016/j.tcs.2024.115062","DOIUrl":"10.1016/j.tcs.2024.115062","url":null,"abstract":"<div><div>We study a popular task model for scheduling parallel real-time tasks, where the internal parallelism of each task is modeled by a directed acyclic graph (DAG). We show that deciding the feasibility of a set of sporadically recurrent DAG tasks is hard for the complexity class <span>PSPACE</span>, thus ruling out approaches to this problem that rely on Integer Linear Programming or Satisfiability solvers (assuming <span><math><mtext>NP</mtext><mo>≠</mo><mtext>PSPACE</mtext></math></span>).</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115062"},"PeriodicalIF":0.9,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143306090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large deviation properties for pattern statistics in primitive rational models","authors":"Massimiliano Goldwurm,&nbsp;Marco Vignati","doi":"10.1016/j.tcs.2024.115055","DOIUrl":"10.1016/j.tcs.2024.115055","url":null,"abstract":"<div><div>We present a large deviation property for pattern statistics representing the number of occurrences of a symbol in words of given length generated at random according to a rational stochastic model. This result is proved assuming that the transition matrix of the model is primitive. We show how the rate function of the large deviation property depends on the main eigenvalues and eigenvectors of the transition matrices associated with the different symbols of the alphabet. We also yield general conditions to guarantee that the range of validity of the large deviation estimate coincides with the whole interval <span><math><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, which represents in our context the largest possible open interval where the property may hold. The case of smaller intervals of validity is finally examined by means of examples.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115055"},"PeriodicalIF":0.9,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143306092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Point enclosure problem for homothetic polygons","authors":"Waseem Akram,&nbsp;Sanjeev Saxena","doi":"10.1016/j.tcs.2024.115054","DOIUrl":"10.1016/j.tcs.2024.115054","url":null,"abstract":"<div><div>In this paper, we investigate the following problem: “given a set <span><math><mi>S</mi></math></span> of <em>n</em> homothetic polygons, preprocess <span><math><mi>S</mi></math></span> to efficiently report all the polygons of <span><math><mi>S</mi></math></span> containing a query point.” A set of polygons is said to be homothetic if each polygon can be obtained from any other polygon in the set using scaling and translation operations. The problem is a counterpart of the homothetic range search problem discussed by Chazelle and Edelsbrunner (1987) <span><span>[9]</span></span>. We show that after preprocessing a set of homothetic polygons with a constant number of vertices, queries can be answered in <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>)</mo></math></span> optimal time, where <em>k</em> is the output size. The preprocessing takes <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> space and time. We also study the problem in a dynamic setting where insertion and deletion operations are allowed. The results we obtain also hold for <em>c</em>-oriented triangles, where <em>c</em> is a fixed constant.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115054"},"PeriodicalIF":0.9,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143305459","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":"An automata-based approach for addressing erroneous behaviors and deadlocks in component-based systems","authors":"Maria Pittou,&nbsp;George Rahonis","doi":"10.1016/j.tcs.2024.115052","DOIUrl":"10.1016/j.tcs.2024.115052","url":null,"abstract":"<div><div>In component-based modeling, a system is built by constructing several coordinating components. In turn, the communication principles among components are modeled by architectures that characterize the permissible interactions and the topology of the components. Several architectures impose also order restrictions to the appearance of the permissible interactions. The application of an architecture to a component-based system does not guarantee that the behaviors of the system meet all the requirements of the architecture. In turn, this can lead the system to actions that do not follow the intended order of the interactions, which we call erroneous behaviors. In this paper, we tackle the problem of such order violations by introducing a formal framework for building component-based systems that avoid erroneous behaviors. For this, we formalize the systems by nondeterministic finite automata and we model their architecture by a fragment of extended propositional interaction logic. This logic achieves to encode the order restrictions in the occurrence of the permissible interactions of the architecture. Given a component-based system with an architecture, we propose an automata-based method for constructing a respective system without erroneous behaviors. Our algorithm returns a system that is also free of deadlocks, meaning that the system does not reach states from which it can no longer progress.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115052"},"PeriodicalIF":0.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143305537","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":"Extensional concepts in intensional type theory, revisited","authors":"Krzysztof Kapulkin,&nbsp;Yufeng Li","doi":"10.1016/j.tcs.2024.115051","DOIUrl":"10.1016/j.tcs.2024.115051","url":null,"abstract":"<div><div>Revisiting a classic result from M. Hofmann's dissertation, we give a direct proof of Morita equivalence, in the sense of V. Isaev, between extensional type theory and intensional type theory extended by the principles of functional extensionality and of uniqueness of identity proofs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115051"},"PeriodicalIF":0.9,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}