Theoretical Computer Science最新文献

{"title":"On the problem of Best Arm Retention","authors":"Houshuang Chen,&nbsp;Yuchen He,&nbsp;Chihao Zhang","doi":"10.1016/j.tcs.2025.115213","DOIUrl":"10.1016/j.tcs.2025.115213","url":null,"abstract":"<div><div>This paper presents a comprehensive study on the problem of Best Arm Retention (BAR), which has recently found applications in streaming algorithms for multi-armed bandits. In the BAR problem, the goal is to retain <em>m</em> arms with the best arm included from <em>n</em> after some trials, in stochastic multi-armed bandit settings. We first investigate pure exploration for the BAR problem under different criteria, and then minimize the regret with specific constraints, in the context of further exploration in streaming algorithms.<ul><li><span>•</span><span><div>We begin by revisiting the lower bound for the <span><math><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-PAC algorithm for Best Arm Identification (BAI) and adapt the classical KL-divergence argument to derive optimal bounds for <span><math><mo>(</mo><mi>ε</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-PAC algorithms for BAR.</div></span></li><li><span>•</span><span><div>We further study another variant of the problem, called <em>r</em>-BAR, which requires the expected gap between the best arm and the optimal arm retained is less than <em>r</em>. We prove tight sample complexity for the problem.</div></span></li><li><span>•</span><span><div>We explore the regret minimization problem for <em>r</em>-BAR and develop algorithm beyond pure exploration. We also propose a conjecture regarding the optimal regret in this setting.</div></span></li></ul></div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1041 ","pages":"Article 115213"},"PeriodicalIF":0.9,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759159","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 improved kernel and parameterized algorithm for deletion to induced matching","authors":"Yuxi Liu,&nbsp;Mingyu Xiao","doi":"10.1016/j.tcs.2025.115215","DOIUrl":"10.1016/j.tcs.2025.115215","url":null,"abstract":"<div><div>A graph is called an induced matching if each vertex in the graph is a degree-1 vertex. The <span>Deletion to Induced Matching</span> problem asks whether we can delete at most <em>k</em> vertices from the input graph such that the remaining graph is an induced matching. This paper studies parameterized algorithms for this problem by taking the size <em>k</em> of the deletion set as the parameter. First, we prove a 6<em>k</em>-vertex kernel for this problem, improving the previous result of 7<em>k</em>. Second, we give an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.6477</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>-time and polynomial-space algorithm, improving the previous running-time bound of <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.7485</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1041 ","pages":"Article 115215"},"PeriodicalIF":0.9,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759158","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":"Existence and construction of a C-shaped module within a floorplan","authors":"Rohit Lohani,&nbsp;Krishnendra Shekhawat","doi":"10.1016/j.tcs.2025.115209","DOIUrl":"10.1016/j.tcs.2025.115209","url":null,"abstract":"<div><div>For a given graph, this paper presents a graph-theoretic approach for creating a floorplan with a specific module, i.e., a C-shaped module. Unlike traditional methods that only consider boundary layouts for floorplan generation, this research considers constraints related to constructing the desired modules. The central objective is to explore how graph theoretic properties can ensure the integration of C-shaped modules within floorplans that have rectangular boundaries.</div><div>A key innovation lies in introducing the concept of non-triviality for these modules, which becomes crucial for achieving the desired non-trivial C-shaped module (a non-trivial module means that it cannot be transformed into other shaped modules by stretching or shrinking its module walls, i.e. if its module walls are stretched or shrinked, then either the bends of its neighboring modules may increase or the given adjacency may not be preserved).</div><div>The proposed solution involves a linear-time algorithm based on the concept of canonical labeling. The algorithm introduces prioritized canonical labeling to generate a non-trivial C-shaped module within the floorplan. It operates on a given plane triangulated graph (PTG) that contains at least one interior <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. The paper outlines the algorithm and establishes the essential conditions for constructing a non-trivial C-shaped module within the floorplan of a given plane triangulated graph (PTG) <em>G</em>. Notably, the algorithm's simplicity and ease of implementation set it apart. In future work, we will focus on generating the existence and construction of other desired shaped modules for the given input graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1040 ","pages":"Article 115209"},"PeriodicalIF":0.9,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747196","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":"The extra local diagnosability and diagnosis algorithm of networks under the PMC model","authors":"Huiling Guo ,&nbsp;Shurong Zhang ,&nbsp;Shanshan Shan ,&nbsp;Lin Chen ,&nbsp;Weihua Yang","doi":"10.1016/j.tcs.2025.115210","DOIUrl":"10.1016/j.tcs.2025.115210","url":null,"abstract":"<div><div>The fault diagnosis has played an important role in the reliability of the large-scale Networks. In practical applications, we do not require global diagnosis and it suffices to determine the working status of some specific vertices (i.e. processors) in the area of the information transmission. Therefore, the sub-network containing the specific vertex <em>v</em> needs to be designed, and then we can judge whether <em>v</em> is faulty after analyzing the syndrome which is obtained by using the diagnosis model in this sub-network. This approach is known as local diagnosis. Since the faulty network should have local connectivity, we consider the <em>h</em>-extra fault model that, after removing all faulty vertices, each component of the network contains more than <em>h</em> vertices. Then we introduce the concept of <em>h</em>-extra local diagnosability and propose a sufficient condition for estimating the extra local diagnosability of the vertex <em>v</em> under the PMC diagnosis model. Additionally, the structure <span><math><mi>T</mi><mo>(</mo><mi>v</mi><mo>;</mo><mi>h</mi><mo>)</mo></math></span> which is the sub-network containing <em>v</em> is designed and we propose the algorithm for correctly identifying the faulty or fault-free status of <em>v</em>. Finally, the simulation demonstrates that, under the PMC model, our diagnosis algorithm is still effective in <span><math><mi>T</mi><mo>(</mo><mi>v</mi><mo>;</mo><mi>h</mi><mo>)</mo></math></span> even with a large number of faulty vertices.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1040 ","pages":"Article 115210"},"PeriodicalIF":0.9,"publicationDate":"2025-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143740042","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":"Integer-valued martingales and cl-Turing reductions","authors":"Nan Fang ,&nbsp;Lu Liu ,&nbsp;Liang Yu","doi":"10.1016/j.tcs.2025.115202","DOIUrl":"10.1016/j.tcs.2025.115202","url":null,"abstract":"<div><div>We prove that a sequence <em>X</em> is integer-valued random (IVR) if and only if for every cl-tt reduction Φ, there are only finitely many oracles that compute <em>X</em> via Φ. A well-known property of integer-valued supermartingales is that under any cone, there is a subcone where the values are constant. We extend this property and prove that for a special integer-valued martingale <em>M</em>, which starts with initial capital 1 and always bets 1 on bit “1” if possible, and for any integer-valued supermartingale <em>f</em>, under any cone there is a subcone such that <em>f</em> actually simulates <em>M</em>. Using this property, we prove that there exists a non-IVR sequence that is cl-computable by only countably many oracles, as a complement to the first result.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1040 ","pages":"Article 115202"},"PeriodicalIF":0.9,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739984","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":"Random expansions of trees with bounded height","authors":"Vera Koponen,&nbsp;Yasmin Tousinejad","doi":"10.1016/j.tcs.2025.115201","DOIUrl":"10.1016/j.tcs.2025.115201","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We consider a sequence &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of trees &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; where, for some &lt;span&gt;&lt;math&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; every &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; has height at most Δ and as &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; the minimal number of children of a nonleaf tends to infinity. We can view every tree as a (first-order) &lt;em&gt;τ&lt;/em&gt;-structure where &lt;em&gt;τ&lt;/em&gt; is a signature with one binary relation symbol. For a fixed (arbitrary) finite and relational signature &lt;span&gt;&lt;math&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;mo&gt;⊇&lt;/mo&gt;&lt;mi&gt;τ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; we consider the set &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of expansions of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; to &lt;em&gt;σ&lt;/em&gt; and a probability distribution &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; which is determined by a (parametrized/lifted) Probabilistic Graphical Model (PGM) &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; which can use the information given by &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;The kind of PGM that we consider uses formulas of a many-valued logic that we call &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; with truth values in the unit interval &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. We also use &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; to express queries, or events, on &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. With this setup we prove that, under some assumptions on &lt;strong&gt;T&lt;/strong&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and a (possibly quite complex) formula &lt;span&gt;&lt;math&gt;&lt;mi&gt;φ&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, as &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are vertices of the tree &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; then ","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1040 ","pages":"Article 115201"},"PeriodicalIF":0.9,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747197","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":"Convex language semantics for nondeterministic probabilistic automata","authors":"Gerco van Heerdt ,&nbsp;Justin Hsu ,&nbsp;Joël Ouaknine ,&nbsp;Alexandra Silva","doi":"10.1016/j.tcs.2025.115191","DOIUrl":"10.1016/j.tcs.2025.115191","url":null,"abstract":"<div><div>We explore language semantics for automata combining probabilistic and nondeterministic behaviors. We first show that there are precisely two natural semantics for probabilistic automata with nondeterminism. For both choices, we show that these automata are strictly more expressive than deterministic probabilistic automata, and we prove that the problem of checking language equivalence is undecidable by reduction from the threshold problem. However, we provide a discounted metric that can be computed to arbitrarily high precision.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1040 ","pages":"Article 115191"},"PeriodicalIF":0.9,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704395","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":"A theory of fine-grained lineage for functions on structured objects","authors":"Sylvain Hallé,&nbsp;Hugo Tremblay","doi":"10.1016/j.tcs.2025.115192","DOIUrl":"10.1016/j.tcs.2025.115192","url":null,"abstract":"<div><div>Lineage is the process of keeping track of the relationship between the inputs of a data processing task and the parts of the output they contribute to produce. Depending on its precise definition, lineage can be seen as a form of database provenance, a means of tracking information flow in computer programs, or be used to express causality and provide counter-examples for the falsity of a logical statement. In this paper, we establish the formal foundations of a notion of lineage for arbitrary abstract functions manipulating objects that are “composite” –that is, can be made of multiple other objects. Three definitions of lineage over functions are formally defined, respectively called explanation, participation and extraction; we then establish explanation relationships for a set of elementary functions, and for compositions thereof. A fully functional implementation of these concepts is finally presented and experimentally evaluated.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1039 ","pages":"Article 115192"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681928","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":"Improved algorithms for optimal k sink location on path networks","authors":"Binay Bhattacharya ,&nbsp;Mordecai J. Golin ,&nbsp;Yuya Higashikawa ,&nbsp;Tsunehiko Kameda ,&nbsp;Naoki Katoh","doi":"10.1016/j.tcs.2025.115190","DOIUrl":"10.1016/j.tcs.2025.115190","url":null,"abstract":"<div><div>We address the problem of placing <em>k</em> sinks on dynamic-flow path networks with <em>n</em> vertices so as to minimize their maximum evacuation completion time. We develop two different algorithms that, when all edges have the same capacity, run respectively in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time. When the edge capacities can be different, i.e., are <em>general</em>, they run respectively in <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>log</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><mi>n</mi><msup><mrow><mi>log</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> time.</div><div>These algorithms improve upon the previously most efficient algorithms, which had time complexities <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mi>n</mi><msup><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>, respectively, for the uniform and general edge capacity models. The improvements are achieved by moving from a dynamic programming based approach to a parametric-search based one.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1039 ","pages":"Article 115190"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696771","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":"Efficient diagonalization of symmetric matrices associated with graphs of small treewidth","authors":"Martin Fürer ,&nbsp;Carlos Hoppen ,&nbsp;Vilmar Trevisan","doi":"10.1016/j.tcs.2025.115187","DOIUrl":"10.1016/j.tcs.2025.115187","url":null,"abstract":"<div><div>Let <span><math><mi>M</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math></span> be a symmetric matrix of order <em>n</em> whose elements lie in an arbitrary field <span><math><mi>F</mi></math></span>, and let <em>G</em> be the graph with vertex set <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span> such that distinct vertices <em>i</em> and <em>j</em> are adjacent if and only if <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn></math></span>. We introduce a dynamic programming algorithm that finds a diagonal matrix that is congruent to <em>M</em>. If <em>G</em> is given with a tree decomposition <span><math><mi>T</mi></math></span> of width <em>k</em>, then this can be done in time <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mo>|</mo><mi>T</mi><mo>|</mo><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>n</mi><mo>)</mo></math></span>, where <span><math><mo>|</mo><mi>T</mi><mo>|</mo></math></span> denotes the number of nodes in <span><math><mi>T</mi></math></span>. Among other things, this allows the computation of the determinant, the rank and the inertia of a symmetric matrix in time <span><math><mi>O</mi><mo>(</mo><mi>k</mi><mo>|</mo><mi>T</mi><mo>|</mo><mo>+</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1040 ","pages":"Article 115187"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143697768","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}