Theoretical Computer Science最新文献

{"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}

{"title":"The distributed algorithms for the lower-bounded k-center clustering in metric space","authors":"Ting Liang ,&nbsp;Xiaoliang Wu ,&nbsp;Jinhui Xu ,&nbsp;Qilong Feng","doi":"10.1016/j.tcs.2024.114975","DOIUrl":"10.1016/j.tcs.2024.114975","url":null,"abstract":"<div><div>Clustering is a fundamental unsupervised machine learning problem. However, due to limited processing memory and CPU power, it is challenging to cluster large-scale data. The distributed methods have received great attention in recent years since large-scale data can be stored and computed on multiple machines. In this paper, we study a variant of the <em>k</em>-center clustering problem, i.e., the lower-bounded <em>k</em>-center clustering problem (denoted as the <span>Lb-</span><em>k</em><span>-Cen</span> problem), in the Massively Parallel Computation (MPC) distributed model. The current best distributed result for the <span>Lb-</span><em>k</em><span>-Cen</span> problem has several rounds of communication between the coordinator and machines, which may increase the local computation and communication cost of the algorithm for handling large-scale data. To achieve fewer local computation and communication rounds, we use the threshold method and flow network technique, which avoid local computation again in each machine, and can achieve a two rounds <span><math><mo>(</mo><mn>9</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></math></span>-approximation algorithm in metric space. Moreover, we also consider the distributed algorithm for the <span>Lb-</span><em>k</em><span>-Cen</span> problem in the metric space with bounded doubling dimension, and propose a two rounds <span><math><mo>(</mo><mn>3</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></math></span>-approximation algorithm.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1027 ","pages":"Article 114975"},"PeriodicalIF":0.9,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138841","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 layered area-proportional rectangle contact representations","authors":"Carolina Haase,&nbsp;Philipp Kindermann","doi":"10.1016/j.tcs.2024.115021","DOIUrl":"10.1016/j.tcs.2024.115021","url":null,"abstract":"<div><div>Semantic word clouds visualize the semantic relatedness between the words of a text by placing pairs of related words close to each other. Formally, the problem of drawing semantic word clouds corresponds to drawing a rectangle contact representation of a graph whose vertices correlate to the words to be displayed and whose edges indicate that two words are semantically related. The goal is to maximize the number of realized contacts while avoiding any false adjacencies. We consider a variant of this problem that restricts input graphs to be layered and all rectangles to be of equal height, called <span>Maximum Layered Contact Representation Of Word Networks</span> or <span>Max-LayeredCrown</span>, as well as the variant <span>Max-IntLayeredCrown</span>, which restricts the problem to only rectangles of integer width and the placement of those rectangles to integer coordinates.</div><div>We classify the corresponding decision problem <em>k</em>-<span>IntLayeredCrown</span> as NP-complete even for internally triangulated planar graphs and <em>k</em>-<span>LayeredCrown</span> as NP-complete for planar graphs. We introduce three algorithms: a 1/2-approximation for <span>Max-LayeredCrown</span> of internally triangulated planar graphs, and a PTAS and an XP algorithm for <span>Max-IntLayeredCrown</span> with rectangle width polynomial in <em>n</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1027 ","pages":"Article 115021"},"PeriodicalIF":0.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138840","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 SARSA and DQN algorithms for reinforcement learning","authors":"Guangyu Yao,&nbsp;Nan Zhang,&nbsp;Zhenhua Duan,&nbsp;Cong Tian","doi":"10.1016/j.tcs.2024.115025","DOIUrl":"10.1016/j.tcs.2024.115025","url":null,"abstract":"<div><div>Reinforcement learning is a branch of machine learning in which an agent interacts with an environment to learn optimal actions that maximize cumulative rewards. This paper aims to enhance the SARSA and DQN algorithms in four key aspects: the <em>ε</em>-greedy policy, reward function, value iteration approach, and sampling probability. The experiments are conducted in three scenarios: path planning, CartPole, and MountainCar. The results show that, in these environments, the improved algorithms exhibit better convergence, higher rewards, and more stable training processes.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1027 ","pages":"Article 115025"},"PeriodicalIF":0.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new fast root-finder for black box polynomials","authors":"Victor Y. Pan ,&nbsp;Soo Go ,&nbsp;Qi Luan ,&nbsp;Liang Zhao","doi":"10.1016/j.tcs.2024.115022","DOIUrl":"10.1016/j.tcs.2024.115022","url":null,"abstract":"<div><div>Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our <em>black box root-finder</em> involves no coefficients and works for a black box polynomial, defined by an oracle (that is, black box subroutine) for its evaluation. Such root-finders have various benefits, e.g., are particularly efficient where a polynomial can be evaluated fast, say, is the sum of a small number of shifted monomials <span><math><msup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mo>)</mo></mrow><mrow><mi>a</mi></mrow></msup></math></span>. With incorporation of a fast algorithm by the first author for compression of a disc on the complex plane without losing roots, our root-finder approximates all <em>d</em> complex zeros of a <em>d</em>th degree polynomial <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> (aka roots of equation <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>) by using <em>near-optimal</em> Las Vegas expected number of bit-operations,<span><span>¹</span></span> that is, the root-finder is expected to run almost as fast as one accesses the coefficients with a precision required for the solution within a prescribed error bound. The only other known near-optimal polynomial root-finder was presented by the first author at ACM STOC 1995. It is quite involved and has never been implemented, while already in its initial implementation our new root-finder competed with user's choice package of root-finding subroutines MPSolve, according to extensive numerical experiments with standard test polynomials. Furthermore we readily extend our black box root-finder to approximation of the <em>eigenvalues of a matrix</em> in record expected bit operation time, while the root-finder of STOC 1995, using the coefficients of <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, does not support such extension.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1027 ","pages":"Article 115022"},"PeriodicalIF":0.9,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The evolutionary dynamics of soft-max policy gradient in multi-agent settings","authors":"Martino Bernasconi ,&nbsp;Federico Cacciamani ,&nbsp;Simone Fioravanti ,&nbsp;Nicola Gatti ,&nbsp;Francesco Trovò","doi":"10.1016/j.tcs.2024.115011","DOIUrl":"10.1016/j.tcs.2024.115011","url":null,"abstract":"<div><div>Policy gradient is one of the most famous algorithms in reinforcement learning. This paper studies the mean dynamics of the <em>soft-max policy gradient algorithm</em> and its properties in multi-agent settings by resorting to evolutionary game theory and dynamical system tools. Unlike most multi-agent reinforcement learning algorithms, whose mean dynamics are a slight variant of the replicator dynamics not affecting the properties of the original dynamics, the soft-max policy gradient dynamics presents a structure significantly different from that of the replicator. In particular, we show that the soft-max policy gradient dynamics in a given game are equivalent to the replicator dynamics in an auxiliary game obtained by a non-convex transformation of the payoffs of the original game. Such a structure gives the dynamics several non-standard properties. The first property we study concerns the convergence to the best response. In particular, while the continuous-time mean dynamics always converge to the best response, the crucial question concerns the convergence speed. Precisely, we show that the space of initializations can be split into two complementary sets such that the trajectories initialized from points of the first set (said <em>good initialization region</em>) directly move to the best response. In contrast, those initialized from points of the second set (said <em>bad initialization region</em>) move first to a series of sub-optimal strategies and then to the best response. Interestingly, in multi-agent adversarial machine learning environments, we show that an adversary can exploit this property to make any current strategy of the learning agent using the soft-max policy gradient fall inside a bad initialization region, thus slowing its learning process and exploiting that policy. When the soft-max policy gradient dynamics is studied in multi-population games, modeling the learning dynamics in self-play, we show that the dynamics preserve the volume of the set of initial points. This property proves that the dynamics cannot converge when the only equilibrium of the game is fully mixed, as the volume of the set of initial points would need to shrink. We also give empirical evidence that the volume expands over time, suggesting that the dynamics in games with fully-mixed equilibrium is chaotic.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1027 ","pages":"Article 115011"},"PeriodicalIF":0.9,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143138844","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":"Self-similarity of communities of the ABCD model","authors":"Jordan Barrett ,&nbsp;Bogumił Kamiński ,&nbsp;Paweł Prałat ,&nbsp;François Théberge","doi":"10.1016/j.tcs.2024.115012","DOIUrl":"10.1016/j.tcs.2024.115012","url":null,"abstract":"<div><div>The <strong>A</strong>rtificial <strong>B</strong>enchmark for <strong>C</strong>ommunity <strong>D</strong>etection (<strong>ABCD</strong>) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known <strong>LFR</strong> model but it is faster and can be investigated analytically. In this paper, we show that the <strong>ABCD</strong> model exhibits some interesting self-similar behaviour, namely, the degree distribution of ground-truth communities is asymptotically the same as the degree distribution of the whole graph (appropriately normalized based on their sizes). As a result, we can not only estimate the number of edges induced by each community but also the number of self-loops and multi-edges generated during the process. Understanding these quantities is important as (a) rewiring self-loops and multi-edges to keep the graph simple is an expensive part of the algorithm, and (b) every rewiring causes the underlying configuration models to deviate slightly from uniform simple graphs on their corresponding degree sequences.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 115012"},"PeriodicalIF":0.9,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178792","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}