Theoretical Computer Science最新文献

{"title":"Probabilistic unifying relations for modelling epistemic and aleatoric uncertainty: Semantics and automated reasoning with theorem proving","authors":"Kangfeng Ye,&nbsp;Jim Woodcock,&nbsp;Simon Foster","doi":"10.1016/j.tcs.2024.114876","DOIUrl":"10.1016/j.tcs.2024.114876","url":null,"abstract":"<div><div>Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant impact on machine intelligence. While many probabilistic algorithms have been used in practice in different domains, their automated verification based on formal semantics is still a relatively new research area. In the last two decades, it has attracted much interest. Many challenges, however, remain. The work presented in this paper, probabilistic unifying relations (ProbURel), takes a step towards our vision to tackle these challenges.</div><div>Our work is based on Hehner's predicative probabilistic programming, but there are several obstacles to the broader adoption of his work. Our contributions here include (1) the formalisation of its syntax and semantics by introducing an Iverson bracket notation to separate relations from arithmetic; (2) the formalisation of relations using Unifying Theories of Programming (UTP) and probabilities outside the brackets using summation over the topological space of the real numbers; (3) the constructive semantics for probabilistic loops using Kleene's fixed-point theorem; (4) the enrichment of its semantics from distributions to subdistributions and superdistributions to deal with the constructive semantics; (5) the unique fixed-point theorem to simplify the reasoning about probabilistic loops; and (6) the mechanisation of our theory in Isabelle/UTP, an implementation of UTP in Isabelle/HOL, for automated reasoning using theorem proving.</div><div>We demonstrate our work with six examples, including problems in robot localisation, classification in machine learning, and the termination of probabilistic loops.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114876"},"PeriodicalIF":0.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524004936/pdfft?md5=ea0cab93ec8117c562627dfa268566b2&pid=1-s2.0-S0304397524004936-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315561","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":"Polynomial-time checking of generalized Sahlqvist syntactic shape","authors":"Krishna B. Manoorkar ,&nbsp;Alessandra Palmigiano ,&nbsp;Mattia Panettiere","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":"1021 ","pages":"Article 114875"},"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":"Bohan Lyu ,&nbsp;Jianzhong Li","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":"1021 ","pages":"Article 114878"},"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":"Zelong Zhao,&nbsp;Nan Zhang,&nbsp;Bin Yu,&nbsp;Zhenhua Duan","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":"1021 ","pages":"Article 114879"},"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":"Cornelius Brand ,&nbsp;Viktoriia Korchemna ,&nbsp;Kirill Simonov ,&nbsp;Michael Skotnica","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":"1021 ","pages":"Article 114877"},"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":"Ionut Cardei,&nbsp;Caner Mutlu,&nbsp;Mihaela Cardei","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":"1020 ","pages":"Article 114871"},"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":"Michela Ascolese ,&nbsp;Andrea Frosini ,&nbsp;Elisa Pergola ,&nbsp;Simone Rinaldi ,&nbsp;Laurent Vuillon","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":"1020 ","pages":"Article 114872"},"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":"Lei Sun ,&nbsp;Zexia Shi ,&nbsp;Jian Liu ,&nbsp;Fang-Wei Fu","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":"1020 ","pages":"Article 114869"},"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":"On recursive tiling of some mathematical objects","authors":"Andrew L. Szilard","doi":"10.1016/j.tcs.2024.114781","DOIUrl":"10.1016/j.tcs.2024.114781","url":null,"abstract":"<div><div>This essay, in a novel way, uses various tiles, such as dominoes, square-, interlocking-, and flow-tiles, to visualize approximations to recursively defined, potentially infinite mathematical objects, such as binary code tables, and variants of the space-filling Hilbert curve. The power of L-systems is shown by defining these tiled objects as L-systems.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1022 ","pages":"Article 114781"},"PeriodicalIF":0.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142699487","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":"Soumen Mandal ,&nbsp;Pranabendu Misra ,&nbsp;Ashutosh Rai ,&nbsp;Saket Saurabh","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":"1021 ","pages":"Article 114870"},"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}