Theoretical Computer Science最新文献

{"title":"Termination of rewriting on reversible Boolean circuits as a free 3-category problem","authors":"Adriano Barile,&nbsp;Stefano Berardi,&nbsp;Luca Roversi","doi":"10.1016/j.tcs.2024.115031","DOIUrl":"10.1016/j.tcs.2024.115031","url":null,"abstract":"<div><div>Reversible Boolean Circuits are an interesting computational model under many aspects and in different fields, ranging from Reversible Computing to Quantum Computing. Our contribution is to describe a specific class of Reversible Boolean Circuits - which is as expressive as classical circuits - as a bi-dimensional diagrammatic programming language. We uniformly represent the Reversible Boolean Circuits we focus on as a free 3-category <strong>Toff</strong>. This formalism allows us to incorporate the representation of circuits and of rewriting rules on them, and to prove termination of rewriting. Termination follows from defining a non-identities-preserving functor from our free 3-category <strong>Toff</strong> into a suitable 3-category <strong>Move</strong> that traces the “moves” applied to wires inside circuits.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115031"},"PeriodicalIF":0.9,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169690","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 FPT approximation scheme and approximate kernel for biclique-free max k-weight SAT: Greedy strikes back","authors":"Pasin Manurangsi","doi":"10.1016/j.tcs.2024.115033","DOIUrl":"10.1016/j.tcs.2024.115033","url":null,"abstract":"<div><div>In the <em>Max k-Weight SAT</em> (aka <em>Max SAT with Cardinality Constraint</em>) problem, we are given a CNF formula with <em>n</em> variables and <em>m</em> clauses together with a positive integer <em>k</em>. The goal is to find an assignment where at most <em>k</em> variables are set to one that satisfies as many constraints as possible. Recently, Jain et al. <span><span>[20]</span></span> gave an FPT approximation scheme (FPT-AS) with running time <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mo>(</mo><mi>d</mi><mi>k</mi><mo>/</mo><mi>ϵ</mi><mo>)</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></msup><mo>⋅</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> for Max <em>k</em>-Weight SAT when the incidence graph is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>d</mi><mo>,</mo><mi>d</mi></mrow></msub></math></span>-free. They asked whether a polynomial-size approximate kernel exists. In this work, we answer this question positively by giving a <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo></math></span>-approximate kernel with <span><math><msup><mrow><mo>(</mo><mfrac><mrow><mi>d</mi><mi>k</mi></mrow><mrow><mi>ϵ</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mi>O</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow></msup></math></span> variables. This also implies an improved FPT-AS with running time <span><math><msup><mrow><mo>(</mo><mi>d</mi><mi>k</mi><mo>/</mo><mi>ϵ</mi><mo>)</mo></mrow><mrow><mi>O</mi><mo>(</mo><mi>d</mi><mi>k</mi><mo>)</mo></mrow></msup><mo>⋅</mo><msup><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo>)</mo></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>. Our approximate kernel is based mainly on a couple of greedy strategies together with a sunflower lemma-style reduction rule.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115033"},"PeriodicalIF":0.9,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169002","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 algorithms for minimum sum vertex cover","authors":"Shubhada Aute ,&nbsp;Fahad Panolan","doi":"10.1016/j.tcs.2024.115032","DOIUrl":"10.1016/j.tcs.2024.115032","url":null,"abstract":"<div><div>A minimum sum vertex cover of an <em>n</em>-vertex graph <em>G</em> is a bijection <span><math><mi>ϕ</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span> that minimizes the cost <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>ϕ</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mi>ϕ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>}</mo></math></span>. Finding a minimum sum vertex cover of a graph (the MSVC problem) is NP-hard. MSVC is studied well in the realm of approximation algorithms. The best-known approximation factor in polynomial time for the problem is 16/9 [Bansal, Batra, Farhadi, and Tetali, SODA 2021]. Recently, Stankovic [APPROX/RANDOM 2022] proved that achieving an approximation ratio better than 1.014 for MSVC is NP-hard, assuming the Unique Games Conjecture. We study the MSVC problem from the perspective of parameterized algorithms. The parameters we consider are the size of a minimum vertex cover and the size of a minimum clique modulator of the input graph. We obtain the following results.<ul><li><span>–</span><span><div>MSVC can be solved in <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> time,</div><div>where <em>k</em> is the size of a minimum vertex cover.</div></span></li><li><span>–</span><span><div>MSVC can be solved in <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>⋅</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> time for some computable function <em>f</em>, where <em>k</em> is the size of a minimum clique modulator.</div></span></li></ul></div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115032"},"PeriodicalIF":0.9,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340467","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":"Faster parameterized algorithm for r-pseudoforest deletion","authors":"Dekel Tsur","doi":"10.1016/j.tcs.2024.115034","DOIUrl":"10.1016/j.tcs.2024.115034","url":null,"abstract":"<div><div>In the <em>r</em><span>-Pseudoforest Deletion</span> problem, the input is a graph <em>G</em> and integers <span><math><mi>k</mi><mo>,</mo><mi>r</mi></math></span>, and the goal is to decide whether there is a set of at most <em>k</em> vertices whose removal from <em>G</em> results in a graph in which every connected component can be made into a tree by deleting at most <em>r</em> edges. In this paper we give an <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mo>(</mo><mn>8</mn><mi>r</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></math></span>-time algorithm for <em>r</em><span>-Pseudoforest Deletion</span> for every <span><math><mi>r</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115034"},"PeriodicalIF":0.9,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143168997","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":"Parameterizing path partitions","authors":"Henning Fernau ,&nbsp;Florent Foucaud ,&nbsp;Kevin Mann ,&nbsp;Utkarsh Padariya ,&nbsp;Rajath Rao K.N.","doi":"10.1016/j.tcs.2024.115029","DOIUrl":"10.1016/j.tcs.2024.115029","url":null,"abstract":"<div><div>We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The <span>Path Partition</span> problem (<span>PP</span>) has been studied extensively, as it includes <span>Hamiltonian Path</span> as a special case. The natural variants where the paths are required to be either <em>induced</em> (<span>Induced Path Partition</span>, <span>IPP</span>) or <em>shortest</em> (<span>Shortest Path Partition</span>, <span>SPP</span>), have received much less attention. Both problems are known to be <span><math><mtext>NP</mtext></math></span>-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that <span>SPP</span> remains <span><math><mtext>NP</mtext></math></span>-hard on undirected bipartite graphs. When parameterized by the natural parameter “number of paths”, both <span>SPP</span> and <span>IPP</span> are shown to be <span><math><mtext>W</mtext><mo>[</mo><mn>1</mn><mo>]</mo></math></span>-hard on DAGs. We also show that SPP is in <span><math><mtext>XP</mtext></math></span> both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (<span>IPP</span> is known to be <span><math><mtext>NP</mtext></math></span>-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in <span><math><mtext>FPT</mtext></math></span>, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of <span>PP</span>. When considering the dual parameterization (graph order minus number of paths), all three variants, <span>IPP</span>, <span>SPP</span> and <span>PP</span>, are shown to be in <span><math><mtext>FPT</mtext></math></span> for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results also transfer to the case of covering by edge-disjoint paths, and purely covering.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115029"},"PeriodicalIF":0.9,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169003","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":"Payment scheduling in the Interval Debt Model","authors":"Tom Friedetzky,&nbsp;David C. Kutner,&nbsp;George B. Mertzios ,&nbsp;Iain A. Stewart,&nbsp;Amitabh Trehan","doi":"10.1016/j.tcs.2024.115028","DOIUrl":"10.1016/j.tcs.2024.115028","url":null,"abstract":"<div><div>The network-based study of financial systems has received considerable attention in recent years but has seldom explicitly incorporated the dynamic aspects of such systems. We consider this problem setting from the temporal point of view and introduce the Interval Debt Model (IDM) and some scheduling problems based on it, namely: <span>Bankruptcy Minimization/Maximization</span>, in which the aim is to produce a payment schedule with at most/at least a given number of bankruptcies; <span>Perfect Scheduling</span>, the special case of the minimization variant where the aim is to produce a schedule with no bankruptcies (that is, a perfect schedule); and <span>Bailout Minimization</span>, in which a financial authority must allocate a smallest possible bailout package to enable a perfect schedule. We show that each of these problems is NP-complete, in many cases even on very restricted input instances. On the positive side, we provide for <span>Perfect Scheduling</span> a polynomial-time algorithm on (rooted) out-trees although in contrast we prove NP-completeness on directed acyclic graphs, as well as on instances with a constant number of nodes (and hence also constant treewidth). When we allow non-integer payments, we show by a linear programming argument that the problem <span>Bailout Minimization</span> can be solved in polynomial time.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115028"},"PeriodicalIF":0.9,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169000","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":"G-good-neighbor diagnosability under the modified comparison model for multiprocessor systems","authors":"Mu-Jiang-Shan Wang ,&nbsp;Dong Xiang ,&nbsp;Sun-Yuan Hsieh","doi":"10.1016/j.tcs.2024.115027","DOIUrl":"10.1016/j.tcs.2024.115027","url":null,"abstract":"<div><div>Diagnosing faults in multiprocessor systems has long been significant due to its performance impact and its blend of Graph Theory and Computer Science concepts. In 2012, Peng et al. introduced the <em>g</em>-good-neighbor diagnosability, ensuring every fault-free node has at least <em>g</em> fault-free neighbors. This concept, gaining traction over the years, has led to extensive research on the connectivity and diagnosability of many prominent multiprocessor systems. In this paper, we introduce a novel comparison model, the MC model, for multiprocessor systems. This model integrates the strengths of both the PMC and MM^⁎ models, optimizing computing power and time. We present an algorithm detailing the MC model's operations and establish the conditions for a multiprocessor system <em>G</em> to be <em>g</em>-good-neighbor <em>t</em>-diagnosable under the MC model. A general method to determine a <em>G</em>'s <em>g</em>-good-neighbor diagnosability under the MC model is also provided. We further highlight the MC model's advantages over the PMC and MM (including MM^⁎) models. Lastly, we apply the MC model to Hypercube, determining its <em>g</em>-good-neighbor diagnosability.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115027"},"PeriodicalIF":0.9,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143168998","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":"Phase transition of the 3-majority opinion dynamics with noisy interactions","authors":"Francesco d'Amore,&nbsp;Isabella Ziccardi","doi":"10.1016/j.tcs.2024.115030","DOIUrl":"10.1016/j.tcs.2024.115030","url":null,"abstract":"<div><div>Communication noise is a common feature in several real-world scenarios where systems of agents need to communicate in order to pursue some collective task. Indeed, many biologically inspired systems that try to achieve agreements on some opinion must implement <em>resilient</em> dynamics, i.e. that are not strongly affected by noisy communications. In this work, we study the <span>3-Majority</span> dynamics, an opinion dynamics that has been shown to be an efficient protocol for the majority consensus problem, in which we introduce a simple feature of uniform communication noise, following D'Amore et al. (2022). We prove that, in the fully connected communication network of <em>n</em> agents and in the binary opinion case, the process induced by the <span>3-Majority</span> dynamics exhibits a phase transition. For a noise probability <span><math><mi>p</mi><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>3</mn></math></span>, the dynamics reach in logarithmic time an almost-consensus metastable phase which lasts for a polynomial number of rounds with high probability. We characterize this phase by showing that there exists an attractive equilibrium value <span><math><msub><mrow><mi>s</mi></mrow><mrow><mtext>eq</mtext></mrow></msub><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span> for the bias of the system, i.e. the difference between the majority community size and the minority one. Moreover, we show that the agreement opinion is the initial majority one if the bias towards it is of magnitude <span><math><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> in the initial configuration. If, instead, <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn><mo>/</mo><mn>3</mn></math></span>, we show that no form of consensus is possible, and any information regarding the initial majority opinion is lost in logarithmic time with high probability. Despite more communications per-round being allowed, the <span>3-Majority</span> dynamics surprisingly turns out to be less resilient to noise than the <span>Undecided-State</span> dynamics, whose noise threshold value is <span><math><mi>p</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115030"},"PeriodicalIF":0.9,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143168999","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":"The complexity of ferromagnetic 2-spin systems on bounded degree graphs","authors":"Zonglei Bai ,&nbsp;Yongzhi Cao ,&nbsp;Hanpin Wang","doi":"10.1016/j.tcs.2024.114940","DOIUrl":"10.1016/j.tcs.2024.114940","url":null,"abstract":"<div><div>Spin systems model the interactions between neighbors on graphs. An important special case is when there are only 2-spins. For 2-spin systems, the problem of approximating the partition function is well understood for anti-ferromagnetic case, while the ferromagnetic case is still not clear. We study the approximability of ferromagnetic 2-spin systems on bounded degree graphs, and make a new step towards the open problem of classifying the ferromagnetic 2-spin systems. On the algorithmic side, we show that the partition function is zero-free for any external field in the whole complex plane except a ring surrounded by two circles with respect to the degree bounds. Especially, for regular graphs, the two circles coincide, and the partition function vanishes only when the external field lies on the circle. Then using Barvinok's method, we obtain a new efficient and deterministic fully polynomial time approximation scheme (FPTAS) for the partition function in the zero-free regions. On the hardness side, we prove the #BIS-hardness of ferromagnetic 2-spin systems on bounded degree graphs. There exists an interval on the real axis so that this problem is #BIS-hard for any external field in the interval. Especially, the upper bound of the interval coincides with the boundary of the zero-free regions, which implies a complexity transition at the point.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 114940"},"PeriodicalIF":0.9,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143168996","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":"FPT approximation for capacitated clustering with outliers","authors":"Rajni Dabas ,&nbsp;Neelima Gupta ,&nbsp;Tanmay Inamdar","doi":"10.1016/j.tcs.2024.115026","DOIUrl":"10.1016/j.tcs.2024.115026","url":null,"abstract":"<div><div>Clustering problems such as <em>k</em>-<span>Median</span>, and <em>k</em>-<span>Means</span>, are motivated from applications such as location planning, unsupervised learning among others. In many such applications, it is important to find the clustering of points that is not “skewed” in terms of the number of points, i.e., no cluster should contain <em>too many</em> points. This is often modeled by introducing <em>capacity constraints</em> on the sizes of clusters. In an orthogonal direction, another important consideration in the domain of clustering is how to handle the presence of <em>outliers</em> in the data. Indeed, the aforementioned clustering problems have been generalized in the literature to separately handle capacity constraints and outliers. However, to the best of our knowledge, there has been very little work on studying the approximability of clustering problems that can simultaneously handle capacity constraints as well as outliers.</div><div>We bridge this gap and initiate the study of the <span>Capacitated</span> <em>k</em><span>-Median with Outliers</span> (<span>C</span><em>k</em><span>MO</span>) problem. In this problem, we want to cluster all except <em>m outlier points</em> into at most <em>k</em> clusters, such that (i) the clusters respect the capacity constraints, and (ii) the cost of clustering, defined as the sum of distances of each <em>non-outlier</em> point to its assigned cluster-center, is minimized.</div><div>We design the first constant-factor approximation algorithms for <span>C</span><em>k</em><span>MO</span>. In particular, our algorithm returns a <span><math><mo>(</mo><mn>3</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></math></span>-approximation for <span>C</span><em>k</em><span>MO</span> in general metric spaces that runs in time <span><math><mi>f</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>ϵ</mi><mo>)</mo><mo>⋅</mo><mo>|</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><msup><mrow><mo>|</mo></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>, where <span><math><mo>|</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>|</mo></math></span> denotes the input size.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1027 ","pages":"Article 115026"},"PeriodicalIF":0.9,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138838","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}