{"title":"The distributed algorithms for the lower-bounded k-center clustering in metric space","authors":"Ting Liang , Xiaoliang Wu , Jinhui Xu , 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, 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, Nan Zhang, Zhenhua Duan, 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 , Soo Go , Qi Luan , 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><sup>1</sup></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 , Federico Cacciamani , Simone Fioravanti , Nicola Gatti , 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}
Jordan Barrett , Bogumił Kamiński , Paweł Prałat , François Théberge
{"title":"Self-similarity of communities of the ABCD model","authors":"Jordan Barrett , Bogumił Kamiński , Paweł Prałat , 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}
{"title":"EFX allocations for indivisible chores: Matching-based approach","authors":"Yusuke Kobayashi , Ryoga Mahara , Souta Sakamoto","doi":"10.1016/j.tcs.2024.115010","DOIUrl":"10.1016/j.tcs.2024.115010","url":null,"abstract":"<div><div>One of the most important topics in discrete fair division is whether an EFX allocation exists for any instance. Although the existence of EFX allocations is a standing open problem for both goods and chores, the understanding of the existence of EFX allocations for chores is less established compared to goods. We study the existence of EFX allocation for chores under the assumption that all agents' cost functions are additive. Specifically, we design polynomial time algorithms for computing EFX allocations for the following three cases: (i) the number of chores is at most twice the number of agents, (ii) the cost functions of all agents except for one induce the same ordering, and (iii) the number of agents is three and each agent has a personalized bi-valued cost function.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 115010"},"PeriodicalIF":0.9,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178791","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}
Venkata Subba Reddy Palagiri , Guru Pratap Sharma , Ismael G. Yero
{"title":"Complexity issues concerning the quadruple Roman domination problem in graphs","authors":"Venkata Subba Reddy Palagiri , Guru Pratap Sharma , Ismael G. Yero","doi":"10.1016/j.tcs.2024.115013","DOIUrl":"10.1016/j.tcs.2024.115013","url":null,"abstract":"<div><div>Given a graph <em>G</em> with vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, a mapping <span><math><mi>h</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>}</mo></math></span> is called a quadruple Roman dominating function (4RDF) for <em>G</em> if it holds the following. Every vertex <em>x</em> such that <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></math></span> satisfies that <span><math><mi>h</mi><mo>(</mo><mi>N</mi><mo>[</mo><mi>x</mi><mo>]</mo><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>N</mi><mo>[</mo><mi>x</mi><mo>]</mo></mrow></msub><mi>h</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≥</mo><mo>|</mo><mo>{</mo><mi>y</mi><mo>:</mo><mi>y</mi><mo>∈</mo><mi>N</mi><mo>(</mo><mi>x</mi><mo>)</mo><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>h</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>≠</mo><mn>0</mn><mo>}</mo><mo>|</mo><mo>+</mo><mn>4</mn></math></span>, where <span><math><mi>N</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>N</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> stands for the open and closed neighborhood of <em>x</em>, respectively. The smallest possible weight <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> among all possible 4RDFs <em>h</em> for <em>G</em> is the quadruple Roman domination number of <em>G</em>, denoted by <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mo>[</mo><mn>4</mn><mi>R</mi><mo>]</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</div><div>This work is focused on complexity aspects for the problem of computing the value of this parameter for several graph classes. Specifically, it is shown that the decision problem concerning <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mo>[</mo><mn>4</mn><mi>R</mi><mo>]</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is NP-complete when restricted to star convex bipartite, comb convex bipartite, split and planar graphs. In contrast, it is also proved that such problem can be efficiently solved for threshold graphs where an exact solution is demonstrated, while for graphs having an efficient dominating set, tight upper and lower bounds in terms of the classical domination number are given. In addition, some approximation results to the problem are given. That is, we show that the problem cannot be approximated within <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>)</mo><mi>ln</mi><mo></mo><mo>|</mo><mi>V</mi><mo>|</mo></math></span> for any <span><math><mi>ϵ</mi><mo>></mo><mn>0</mn></math></span> unless <span><math><mi>P</mi><mo>=</mo><mi>N</mi><mi>P</mi></math></span>.","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 115013"},"PeriodicalIF":0.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178839","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":"Egalitarian roommate allocations: Complexity and stability","authors":"Vincenzo Bonifaci , Helena Rivera Dallorto","doi":"10.1016/j.tcs.2024.115009","DOIUrl":"10.1016/j.tcs.2024.115009","url":null,"abstract":"<div><div>We study two roommate assignment problems, called Ordinal Roommate Allocation and Cardinal Roommate Allocation, where students have preferences over roommates, rooms have varying capacities, and the goal is to maximize the minimum payoff of the students (under two distinct notions of payoff). Both problems are <span><math><mi>NP</mi></math></span>-hard when room sizes are unrestricted. In contrast, the Ordinal Roommate Allocation problem becomes tractable when the maximum room capacity is fixed, while the Cardinal Roommate Allocation problem remains <span><math><mi>NP</mi></math></span>-hard even with bounded room capacity and number of preferences. We then analyze the problems through the lens of stability, considering envy-freeness and a weaker notion we call swap-resistance. Not all instances guarantee an envy-free outcome, and it is shown to be <span><math><mi>NP</mi></math></span>-hard to determine which ones do. However, swap-resistance is always achievable using an efficient algorithm. We discuss connections and distinctions between our work and existing research about utilitarian matchings and stable roommate problems.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 115009"},"PeriodicalIF":0.9,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178790","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":"Exact and inexact search for 2d side-sharing tandems","authors":"Shoshana Marcus , Dina Sokol , Sarah Zelikovitz","doi":"10.1016/j.tcs.2024.115005","DOIUrl":"10.1016/j.tcs.2024.115005","url":null,"abstract":"<div><div>A side-sharing tandem is a rectangular array that is composed of two adjacent non-overlapping occurrences of the same rectangular block. Furthering our understanding of side-sharing tandems can facilitate the development of more efficient 2d pattern matching techniques and may lead to improvements in multi-dimensional compression schemes. Existing algorithms for finding side-sharing tandems are far from optimal on 2d arrays that contain relatively few side-sharing tandems. In this paper, we introduce the idea of a run of side-sharing tandems, as a maximally extended chain of 2d tandems. We demonstrate tight upper bounds on the number of runs of side-sharing tandems that can occur in a rectangular array. We develop an algorithm that locates all <em>τ</em> runs of side-sharing tandems in an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> input array in <span><math><mi>O</mi><mo>(</mo><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>τ</mi><mo>)</mo><mi>log</mi><mo></mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> time. We also introduce several versions of approximate side-sharing tandems with <em>k</em> mismatches along with efficient algorithms for locating them in a rectangular array.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 115005"},"PeriodicalIF":0.9,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178840","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}
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