Theoretical Computer Science最新文献

{"title":"Edge open packing: Complexity, algorithmic aspects, and bounds","authors":"Boštjan Brešar ,&nbsp;Babak Samadi","doi":"10.1016/j.tcs.2024.114884","DOIUrl":"10.1016/j.tcs.2024.114884","url":null,"abstract":"<div><div>Given a graph <em>G</em>, two edges <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> are said to have a common edge <span><math><mi>e</mi><mo>≠</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> if <em>e</em> joins an endvertex of <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> to an endvertex of <span><math><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. A subset <span><math><mi>B</mi><mo>⊆</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is an edge open packing set in <em>G</em> if no two edges of <em>B</em> have a common edge in <em>G</em>, and the maximum cardinality of such a set in <em>G</em> is called the edge open packing number, <span><math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of <em>G</em>. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree 4, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper (Chelladurai et al. (2022) <span><span>[5]</span></span>). Notably, we characterize the graphs <em>G</em> that attain the upper bound <span><math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>/</mo><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1022 ","pages":"Article 114884"},"PeriodicalIF":0.9,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322926","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":"Geometrical Penrose tilings are characterized by their 1-atlas","authors":"Thomas Fernique ,&nbsp;Victor Lutfalla","doi":"10.1016/j.tcs.2024.114883","DOIUrl":"10.1016/j.tcs.2024.114883","url":null,"abstract":"<div><div>Penrose rhombus tilings are tilings of the plane by two decorated rhombi such that the decorations match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove the decorations, we get, by definition, a sofic tiling space that we here call geometrical Penrose tilings. Here, we show how to compute the patterns of a given size which appear in these tilings by three different methods: two based on the substitutive structure of the Penrose tilings and the last on their definition by the cut and projection method. We use this to prove that the geometrical Penrose tilings are characterized by a small set of patterns called vertex-atlas, <em>i.e.</em>, they form a tiling space of finite type. Though considered as folklore, no complete proof of this result has been published, to our knowledge.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1022 ","pages":"Article 114883"},"PeriodicalIF":0.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318871","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":"Arbitrary pattern formation on a continuous circle by oblivious robot swarm","authors":"Brati Mondal,&nbsp;Pritam Goswami,&nbsp;Avisek Sharma,&nbsp;Buddhadeb Sau","doi":"10.1016/j.tcs.2024.114882","DOIUrl":"10.1016/j.tcs.2024.114882","url":null,"abstract":"<div><div>In the field of distributed systems, the Arbitrary Pattern Formation (APF) problem is an extensively studied problem. The purpose of APF is to design an algorithm to move a swarm of robots to a particular position in an environment (discrete or continuous) such that the swarm can form a specific but arbitrary pattern given previously to every robot as an input. In this paper, the solvability of the APF problem on a continuous circle is discussed for a swarm of oblivious and silent robots without chirality under a semi-synchronous scheduler. Firstly a class of configurations (the initial placements of the robots on the circle) called <em>Formable Configuration</em> (<em>FC</em>) has been provided which is necessary to solve the APF problem on a continuous circle. Then considering the initial configuration to be an <em>FC</em>, a deterministic and distributed algorithm has been provided that solves the APF problem for <em>n</em> robots on a continuous circle of fixed radius within <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> epochs without collision, where an epoch is considered to be a time interval in which all robots are activated at least once.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114882"},"PeriodicalIF":0.9,"publicationDate":"2024-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315560","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":"L(3,2,1)-labeling of certain planar graphs","authors":"Tiziana Calamoneri","doi":"10.1016/j.tcs.2024.114881","DOIUrl":"10.1016/j.tcs.2024.114881","url":null,"abstract":"<div><div>Given a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> of maximum degree <em>Δ</em>, denoting by <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> the distance in <em>G</em> between nodes <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>V</mi></math></span>, an <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-labeling of <em>G</em> is an assignment <em>l</em> from <em>V</em> to the set of non-negative integers such that <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>3</mn></math></span> if <em>x</em> and <em>y</em> are adjacent, <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>2</mn></math></span> if <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>, and <span><math><mo>|</mo><mi>l</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>l</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mn>1</mn></math></span> if <span><math><mi>d</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>3</mn></math></span>, for all <em>x</em> and <em>y</em> in <em>V</em>. The <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number <span><math><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the smallest positive integer such that <em>G</em> admits an <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-labeling with labels from <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>λ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>}</mo></math></span>.</div><div>In this paper, the <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number of certain planar graphs is determined, proving that it is linear in <em>Δ</em>, although the general upper bound for the <span><math><mi>L</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-number of planar graphs is quadratic in <em>Δ</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1022 ","pages":"Article 114881"},"PeriodicalIF":0.9,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318691","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":"Complexity and enumeration in models of genome rearrangement","authors":"Lora Bailey ,&nbsp;Heather Smith Blake ,&nbsp;Garner Cochran ,&nbsp;Nathan Fox ,&nbsp;Michael Levet ,&nbsp;Reem Mahmoud ,&nbsp;Elizabeth Bailey Matson ,&nbsp;Inne Singgih ,&nbsp;Grace Stadnyk ,&nbsp;Xinyi Wang ,&nbsp;Alexander Wiedemann","doi":"10.1016/j.tcs.2024.114880","DOIUrl":"10.1016/j.tcs.2024.114880","url":null,"abstract":"<div><div>In this paper, we examine the computational complexity of enumeration in certain genome rearrangement models. We first show that the <span>Pairwise Rearrangement</span> problem in the Single Cut-and-Join model (Bergeron et al., 2010 <span><span>[8]</span></span>) is <span><math><mi>#</mi><mtext>P</mtext></math></span>-complete under polynomial-time Turing reductions. Next, we show that in the Single Cut or Join model (Feijão and Meidanis, 2011 <span><span>[21]</span></span>), the problem of enumerating all medians (<figure><img></figure>) is logspace-computable (<span><math><mtext>FL</mtext></math></span>), improving upon the previous polynomial-time (<span><math><mtext>FP</mtext></math></span>) bound of Miklós &amp; Smith <span><span>[41]</span></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1022 ","pages":"Article 114880"},"PeriodicalIF":0.9,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318872","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":"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}