Theoretical Computer Science最新文献

{"title":"Polynomial-time checking of generalized Sahlqvist syntactic shape","authors":"","doi":"10.1016/j.tcs.2024.114875","DOIUrl":"10.1016/j.tcs.2024.114875","url":null,"abstract":"<div><p>The best known modal logics are axiomatized by Sahlqvist axioms, i.e., axioms of a syntactic shape which guarantees these formulas to have such excellent properties as canonicity and elementarity. Recently, the definition of Sahlqvist formulas has been generalized and extended from formulas in classical modal logic to inequalities (sequents) in a wide family of logics known as LE-logics. We introduce an algorithm which checks if a given inequality is generalized Sahlqvist in polynomial time.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524004924/pdfft?md5=45608c7f83fcb56514df88d27c390c23&pid=1-s2.0-S0304397524004924-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271274","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":"Multiple-model polynomial regression and efficient algorithms for data analysis","authors":"","doi":"10.1016/j.tcs.2024.114878","DOIUrl":"10.1016/j.tcs.2024.114878","url":null,"abstract":"<div><p>This paper newly proposes a data analysis method using multiple-model <em>p</em>-order polynomial regression (MMPR), which separates given datasets into subsets and constructs respective polynomial regression models for them. An approximate algorithm to construct MMPR models based on <span><math><mo>(</mo><mi>ϵ</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-estimator, and mathematical proofs of the correctness and efficiency of the algorithm are introduced. This paper empirically implements the method on both synthetic and real-world datasets, and it's shown to have comparable performance to existing regression methods in many cases, while it takes almost the shortest time to provide a regression model with high prediction accuracy.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271276","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":"Generating Java code pairing with ChatGPT","authors":"","doi":"10.1016/j.tcs.2024.114879","DOIUrl":"10.1016/j.tcs.2024.114879","url":null,"abstract":"<div><p>The Large Language Models (LLMs) like ChatGPT 3.5 have created a new era of automatic code generation. However, the existing research primarily focuses on generating simple code based on datasets (such as HumanEval, etc.). Most of approaches pay less attention to complex and practical code generation. Therefore, in this paper, we propose a new approach called “Xd-CodeGen” which can be used to generate large scale Java code. This approach is composed of four phases: requirement analysis, modeling, code generation, and code verification. In the requirement analysis phase, ChatGPT 3.5 is utilized to decompose and restate user requirements. To do so, a knowledge graph is developed to describe entities and their relationship in detail. Further, Propositional Projection Temporal Logic (PPTL) formulas are employed to define the properties of requirements. In the modeling phase, we use knowledge graphs to enhance prompts and generate UML class and activity diagrams for each sub-requirement using ChatGPT 3.5. In the code generation phase, based on established UML models, we make use of prompt engineering and knowledge graph to generate Java code. In the code verification phase, a runtime verification at code level approach is employed to verify generated Java code. Finally, we apply the proposed approach to develop a practical Java web project.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271275","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":"Counting vanishing matrix-vector products","authors":"","doi":"10.1016/j.tcs.2024.114877","DOIUrl":"10.1016/j.tcs.2024.114877","url":null,"abstract":"<div><p>Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let <span><math><mi>v</mi><mo>∈</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> be a rational vector, <span><math><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span> a list of <span><math><mi>d</mi><mo>×</mo><mi>d</mi></math></span> rational matrices, <span><math><mi>S</mi><mo>∈</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>h</mi><mo>×</mo><mi>d</mi></mrow></msup></math></span> a rational matrix not necessarily square and <em>k</em> a parameter. The goal is to compute the number of ways one can choose <em>k</em> matrices <span><math><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math></span> from the list such that <span><math><mi>S</mi><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>⋯</mo><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mi>v</mi><mo>=</mo><mn>0</mn><mo>∈</mo><msup><mrow><mi>Q</mi></mrow><mrow><mi>h</mi></mrow></msup></math></span>.</p><p>In this paper, we show that this problem is <span><math><mi>#</mi><mi>W</mi><mo>[</mo><mn>2</mn><mo>]</mo></math></span>-hard for parameter <em>k</em>. As a consequence, computing the <em>k</em>-th homotopy group of a <em>d</em>-dimensional 1-connected topological space for <span><math><mi>d</mi><mo>&gt;</mo><mn>3</mn></math></span> is <span><math><mi>#</mi><mi>W</mi><mo>[</mo><mn>2</mn><mo>]</mo></math></span>-hard for parameter <em>k</em>. We also discuss a decision version of the problem and its several modifications for which we show <span><math><mi>W</mi><mo>[</mo><mn>1</mn><mo>]</mo><mo>/</mo><mi>W</mi><mo>[</mo><mn>2</mn><mo>]</mo></math></span>-hardness. This is in contrast to the parameterized <em>k</em>-sum problem, which is only <span><math><mi>W</mi><mo>[</mo><mn>1</mn><mo>]</mo></math></span>-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524004948/pdfft?md5=2c53a58210c875914cfbec0e130e8130&pid=1-s2.0-S0304397524004948-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142244070","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":"Space-time graph path planner for unsignalized intersection management with a V2V agent coordination architecture","authors":"","doi":"10.1016/j.tcs.2024.114871","DOIUrl":"10.1016/j.tcs.2024.114871","url":null,"abstract":"<div><p>Reducing traffic congestion and increasing passenger safety are important objectives for emerging automated transportation systems. Autonomous intersection management systems (AIMS) enable large scale optimization of vehicle trajectories with connected and autonomous vehicles (CAVs). We propose a novel approach for computing the fastest waypoint trajectory in intersections using graph search in a discretized space-time graph that produces collision-free paths with variable vehicle speeds that comply with traffic rules and vehicle dynamical constraints. To assist our planner algorithm in decentralized scenarios, we also propose a multi-agent protocol architecture for vehicle coordination for trajectory planning using a vehicle-to-vehicle (V2V) network. The trajectories generated allow a much higher evacuation rate and congestion threshold, with lower <span><math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math></span> algorithm runtime compared to the state of the art conflict detection graph platoon path planning method, even for large scenarios with vehicle arrival rate of <span><math><mn>1</mn><mo>/</mo><mi>s</mi></math></span> and thousands of vehicles.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142239252","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 algebraic approach to the reconstruction of uniform hypergraphs from their degree sequence","authors":"","doi":"10.1016/j.tcs.2024.114872","DOIUrl":"10.1016/j.tcs.2024.114872","url":null,"abstract":"<div><p>The reconstruction of a (hyper)graph starting from its degree sequence is one of the most relevant among the inverse problems investigated in the field of graph theory. In case of graphs, a feasible solution can be quickly reached, while in case of hypergraphs Deza et al. (2018) proved that the problem is NP-hard even in the simple case of 3-uniform ones. This result opened a new research line consisting in the detection of instances for which a solution can be computed in polynomial time. In this work we deal with 3-uniform hypergraphs, and we study them from a different perspective, exploiting a connection of these objects with partially ordered sets. More precisely, we introduce a simple partially ordered set, whose ideals are in bijection with a subclass of 3-uniform hypergraphs. We completely characterize their degree sequences in case of principal ideals (here a simple fast reconstruction strategy follows), and we furthermore carry on a complete analysis of those degree sequences related to the ideals with two generators. We also consider unique hypergraphs in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>e</mi><mi>x</mi><mi>t</mi></mrow></msup></math></span>, i.e., those hypergraphs that do not share their degree sequence with other non-isomorphic ones. We show that uniqueness holds in case of hypergraphs associated to principal ideals, and we provide some examples of hypergraphs in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>e</mi><mi>x</mi><mi>t</mi></mrow></msup></math></span> where this property is lost.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524004894/pdfft?md5=9a5013316177a513ac9292e891e55cf0&pid=1-s2.0-S0304397524004894-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142239250","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":"Constructions of rotation symmetric Boolean functions satisfying almost all cryptographic criteria","authors":"","doi":"10.1016/j.tcs.2024.114869","DOIUrl":"10.1016/j.tcs.2024.114869","url":null,"abstract":"<div><p>Constructions of Boolean functions with various cryptographic properties have always been an important challenge in cryptography. This paper proposes systematic constructions of even-variable rotation symmetric Boolean functions satisfying almost all cryptographic criteria, that is, resiliency, optimal algebraic degree, strict avalanche criterion, high nonlinearity, nonexistence of nonzero linear structures, good global avalanche characteristics. Moreover, some of the constructions also have high algebraic immunity. This is the first time that Boolean functions having such cryptographic properties are obtained, which can be considered as good candidates for the design of real-life encryption schemes.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142239253","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":"Parameterized approximation algorithms for weighted vertex cover","authors":"","doi":"10.1016/j.tcs.2024.114870","DOIUrl":"10.1016/j.tcs.2024.114870","url":null,"abstract":"&lt;div&gt;&lt;p&gt;A &lt;em&gt;vertex cover&lt;/em&gt; of a graph is a set of vertices of the graph such that every edge has at least one endpoint in it. In this work, we study &lt;span&gt;Weighted Vertex Cover&lt;/span&gt; with solution size as a parameter. Formally, in the &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-&lt;span&gt;Vertex Cover&lt;/span&gt; problem, given a graph &lt;em&gt;G&lt;/em&gt;, an integer &lt;em&gt;k&lt;/em&gt;, a positive rational &lt;em&gt;W&lt;/em&gt;, and a weight function &lt;span&gt;&lt;math&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Q&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;, the question is whether &lt;em&gt;G&lt;/em&gt; has a vertex cover of size at most &lt;em&gt;k&lt;/em&gt; of weight at most &lt;em&gt;W&lt;/em&gt;, with &lt;em&gt;k&lt;/em&gt; being the parameter. An &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-bi-criteria approximation algorithm for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-&lt;span&gt;Vertex Cover&lt;/span&gt; either produces a vertex cover &lt;em&gt;S&lt;/em&gt; such that &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, or decides that there is no vertex cover of size at most &lt;em&gt;k&lt;/em&gt; of weight at most &lt;em&gt;W&lt;/em&gt;. We obtain the following results.&lt;/p&gt;&lt;ul&gt;&lt;li&gt;&lt;span&gt;•&lt;/span&gt;&lt;span&gt;&lt;p&gt;A simple &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-bi-criteria approximation algorithm for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-&lt;span&gt;Vertex Cover&lt;/span&gt; in polynomial time by modifying the standard &lt;span&gt;LP&lt;/span&gt;-rounding algorithm.&lt;/p&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;•&lt;/span&gt;&lt;span&gt;&lt;p&gt;A simple exact parameterized algorithm for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-&lt;span&gt;Vertex Cover&lt;/span&gt; running in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&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;msup&gt;&lt;mrow&gt;&lt;mn&gt;1.4656&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; time&lt;span&gt;&lt;span&gt;&lt;sup&gt;1&lt;/sup&gt;&lt;/span&gt;&lt;/span&gt;.&lt;/p&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;•&lt;/span&gt;&lt;span&gt;&lt;p&gt;A &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-approximation algorithm for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-&lt;span&gt;Vertex Cover&lt;/span&gt; running in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&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;msup&gt;&lt;mrow&gt;&lt;mn&gt;1.4656&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;ϵ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; time.&lt;/p&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;•&lt;/span&gt;&lt;span&gt;&lt;p&gt;A &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1.5&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1.5&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-approximation algorithm for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-&lt;span&gt;Vertex Cover&lt;/span&gt; running in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;O&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;msup&gt;&lt;mrow&gt;&lt;mn&gt;1.414&lt;/mn&gt;&lt;/mr","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142271415","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 3-server on a circle and the limitation of canonical potentials","authors":"","doi":"10.1016/j.tcs.2024.114844","DOIUrl":"10.1016/j.tcs.2024.114844","url":null,"abstract":"<div><p>The deterministic <em>k</em>-server conjecture states that there is a <em>k</em>-competitive deterministic algorithm for the <em>k</em>-server problem for any metric space. We show that the work function algorithm is 3-competitive for the 3-server problem on circle metrics, a case left open by Coester and Koutsoupias (2021). Our analysis follows the existing framework but introduces a new potential function which may be viewed as a relaxation of the counterpart by Coester and Koutsoupias (2021). We further notice that the new potential function and many existing ones can be rewritten in a canonical form. Through a computer-aided verification, however, we find that no such canonical potential function can resolve the deterministic 3-server conjecture for general metric spaces under the current analysis framework.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142239251","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":"Approximations for the Steiner Multicycle problem","authors":"","doi":"10.1016/j.tcs.2024.114836","DOIUrl":"10.1016/j.tcs.2024.114836","url":null,"abstract":"<div><p>The <span>Steiner Multicycle</span> problem consists in, given a complete graph, a weight function on its edges, and a collection of pairwise disjoint non-unitary sets called terminal sets, finding a minimum weight collection of vertex-disjoint cycles in the graph such that, for every terminal set, all of its vertices are in a same cycle of the collection. This problem generalizes the <span>Traveling Salesman</span> problem and, therefore, is hard to approximate in general. On the practical side, it models a collaborative less-than-truckload problem with pickup and delivery locations. Using an algorithm for the <span>Survivable Network Design</span> problem and <em>T</em>-joins, we obtain a 3-approximation for the metric case, improving on the previous best 4-approximation. Furthermore, we present an (11/9)-approximation for the particular case of the <span>Steiner Multicycle</span> in which each edge weight is 1 or 2. This algorithm can be adapted to obtain a (7/6)-approximation when every terminal set contains at least four vertices. Finally, we devise an <span><math><mi>O</mi><mo>(</mo><mi>lg</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-approximation algorithm for the asymmetric version of the problem.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163957","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}