Theoretical Computer Science最新文献

{"title":"Point enclosure problem for homothetic polygons","authors":"Waseem Akram,&nbsp;Sanjeev Saxena","doi":"10.1016/j.tcs.2024.115054","DOIUrl":"10.1016/j.tcs.2024.115054","url":null,"abstract":"<div><div>In this paper, we investigate the following problem: “given a set <span><math><mi>S</mi></math></span> of <em>n</em> homothetic polygons, preprocess <span><math><mi>S</mi></math></span> to efficiently report all the polygons of <span><math><mi>S</mi></math></span> containing a query point.” A set of polygons is said to be homothetic if each polygon can be obtained from any other polygon in the set using scaling and translation operations. The problem is a counterpart of the homothetic range search problem discussed by Chazelle and Edelsbrunner (1987) <span><span>[9]</span></span>. We show that after preprocessing a set of homothetic polygons with a constant number of vertices, queries can be answered in <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo>)</mo></math></span> optimal time, where <em>k</em> is the output size. The preprocessing takes <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> space and time. We also study the problem in a dynamic setting where insertion and deletion operations are allowed. The results we obtain also hold for <em>c</em>-oriented triangles, where <em>c</em> is a fixed constant.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115054"},"PeriodicalIF":0.9,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143305459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An automata-based approach for addressing erroneous behaviors and deadlocks in component-based systems","authors":"Maria Pittou,&nbsp;George Rahonis","doi":"10.1016/j.tcs.2024.115052","DOIUrl":"10.1016/j.tcs.2024.115052","url":null,"abstract":"<div><div>In component-based modeling, a system is built by constructing several coordinating components. In turn, the communication principles among components are modeled by architectures that characterize the permissible interactions and the topology of the components. Several architectures impose also order restrictions to the appearance of the permissible interactions. The application of an architecture to a component-based system does not guarantee that the behaviors of the system meet all the requirements of the architecture. In turn, this can lead the system to actions that do not follow the intended order of the interactions, which we call erroneous behaviors. In this paper, we tackle the problem of such order violations by introducing a formal framework for building component-based systems that avoid erroneous behaviors. For this, we formalize the systems by nondeterministic finite automata and we model their architecture by a fragment of extended propositional interaction logic. This logic achieves to encode the order restrictions in the occurrence of the permissible interactions of the architecture. Given a component-based system with an architecture, we propose an automata-based method for constructing a respective system without erroneous behaviors. Our algorithm returns a system that is also free of deadlocks, meaning that the system does not reach states from which it can no longer progress.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115052"},"PeriodicalIF":0.9,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143305537","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":"Extensional concepts in intensional type theory, revisited","authors":"Krzysztof Kapulkin,&nbsp;Yufeng Li","doi":"10.1016/j.tcs.2024.115051","DOIUrl":"10.1016/j.tcs.2024.115051","url":null,"abstract":"<div><div>Revisiting a classic result from M. Hofmann's dissertation, we give a direct proof of Morita equivalence, in the sense of V. Isaev, between extensional type theory and intensional type theory extended by the principles of functional extensionality and of uniqueness of identity proofs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115051"},"PeriodicalIF":0.9,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340393","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":"On the complexity of minimizing energy consumption of partitioning DAG tasks","authors":"Wei Liu ,&nbsp;Jian-Jia Chen ,&nbsp;Yongjie Yang","doi":"10.1016/j.tcs.2024.115047","DOIUrl":"10.1016/j.tcs.2024.115047","url":null,"abstract":"<div><div>We study a graph partition problem where the input is a directed acyclic graph (DAG) representing tasks as vertices and dependencies between tasks as arcs. The goal is to assign the tasks to <em>k</em> heterogeneous machines in a way that minimizes the total energy consumed for completing the tasks. We first show that the problem is <span>NP</span>-hard. Then, we present polynomial-time algorithms for two special cases: one where there are only two machines, and another where the input DAG is a directed path. Finally, we examine a variant where there are only two machines, with one capable of executing a limited number of tasks, and demonstrate that this special case remains computationally hard.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1030 ","pages":"Article 115047"},"PeriodicalIF":0.9,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143306091","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":"Identity-based matchmaking encryption with stronger security and instantiation on lattices","authors":"Yuejun Wang ,&nbsp;Baocang Wang ,&nbsp;Qiqi Lai ,&nbsp;Yu Zhan","doi":"10.1016/j.tcs.2024.115048","DOIUrl":"10.1016/j.tcs.2024.115048","url":null,"abstract":"<div><div>An identity-based matchmaking encryption (IB-ME) scheme proposed at JOC 2021 supports anonymous but authenticated communications in a way that communication parties can both specify the senders or receivers on the fly. IB-ME is easy to be used in several network applications requiring privacy-preserving for its efficient implementation and special syntax. Despite the rigorous security proofs in previous security models, the existing IB-ME schemes are still possibly vulnerable to some potential neglected attacks. Aiming at the above problems, we provide a stronger security definition of authenticity considering new attacks to fit real-world scenarios and then propose a generic construction of IB-ME satisfying the new model. Inspired by the prior IB-ME construction of Chen et al., the proposed scheme is constructed by combining 2-level anonymous hierarchical IBE (HIBE) and identity-based signature (IBS) schemes. In order to upgrade lattice-based IB-ME with better efficiency, we additionally improve a lattice IBS, as an independent technical contribution, to shorten its signature and thus reduce the final IB-ME ciphertext size. By combining the improved IBS and any 2-level adaptively-secure lattice-based HIBE with anonymity, we finally obtain the <em>first</em> lattice-based IB-ME construction achieving privacy and new-proposed stronger authenticity simultaneously.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115048"},"PeriodicalIF":0.9,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340392","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":"Approximation algorithms for cycle and path partitions in complete graphs","authors":"Jingyang Zhao,&nbsp;Mingyu Xiao","doi":"10.1016/j.tcs.2024.115049","DOIUrl":"10.1016/j.tcs.2024.115049","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Given an edge-weighted (metric/general) complete graph with &lt;em&gt;n&lt;/em&gt; vertices, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, the maximum weight (metric/general) &lt;em&gt;k&lt;/em&gt;-cycle/path partition problem is to find a set of &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; vertex-disjoint &lt;em&gt;k&lt;/em&gt;-cycles/paths such that the total weight is maximized. In this paper, we consider approximation algorithms. For metric &lt;em&gt;k&lt;/em&gt;-cycle partition, we improve the previous approximation ratio from &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, and from &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for constant odd &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and to &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for even &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. For metric &lt;em&gt;k&lt;/em&gt;-path partition, we improve the approximation ratio from &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;27&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;48&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;32&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;36&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;24&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;8&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;10&lt;/mn&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. For the case of &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we improve the approximation ratio from &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; to &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; for metric 4-cycle partition, from &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;m","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115049"},"PeriodicalIF":0.9,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340394","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 strongly polynomial time approximation algorithm for the min-max clustered cycle cover problem","authors":"Pengxiang Pan ,&nbsp;Hongtao Zhu","doi":"10.1016/j.tcs.2024.115050","DOIUrl":"10.1016/j.tcs.2024.115050","url":null,"abstract":"<div><div>We reconsider the min-max clustered cycle cover (MM-CCC) problem, which is described as follows. Given an undirected complete graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>;</mo><mi>w</mi><mo>)</mo></math></span> with a positive integer <em>k</em>, where the vertex set <em>V</em> is partitioned into <em>h</em> clusters <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>h</mi></mrow></msub></math></span>, and <span><math><mi>w</mi><mo>:</mo><mi>E</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> is an edge-weight function satisfying the triangle inequality, it is asked to find <em>k</em> cycles such that they traverse all vertices and the vertices in each cluster are required to be traversed consecutively. The objective is to minimize the weight of the maximum weight cycle. We propose a strongly polynomial time 16-approximation algorithm for the MM-CCC problem. The result improves the previous algorithm in terms of running time.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115050"},"PeriodicalIF":0.9,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340391","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":"Super spanning connectivity of the generalized hypercube network","authors":"Xiaoqian Wang,&nbsp;Eminjan Sabir","doi":"10.1016/j.tcs.2024.115038","DOIUrl":"10.1016/j.tcs.2024.115038","url":null,"abstract":"<div><div>The generalized hypercube <span><math><mi>G</mi><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> is one of the key interconnection networks with attractive topological properties. In this paper, we focus our attention on the super spanning connectivity of <span><math><mi>G</mi><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>. We show that for a pair of arbitrary nodes <em>x</em> and <em>y</em> in <span><math><mi>G</mi><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mspace></mspace><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mn>3</mn><mo>,</mo><mspace></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, there is a set of <span><math><mi>s</mi><mspace></mspace><mo>(</mo><mn>1</mn><mo>≤</mo><mi>s</mi><mo>≤</mo><mi>κ</mi><mo>(</mo><mi>G</mi><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>)</mo><mo>)</mo></math></span> internally node-disjoint <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span>-paths whose union covers every vertex in <span><math><mi>G</mi><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><mi>κ</mi><mo>(</mo><mi>G</mi><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>)</mo></math></span> denotes the connectivity of <span><math><mi>G</mi><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>. Our results, in some sense, extended a previous result in Shih and Kao (2011) <span><span>[23]</span></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115038"},"PeriodicalIF":0.9,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340469","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":"Highly irregular graph decompositions","authors":"Julien Bensmail ,&nbsp;Malory Marin ,&nbsp;Leandro Montero ,&nbsp;Alexandre Talon","doi":"10.1016/j.tcs.2024.115036","DOIUrl":"10.1016/j.tcs.2024.115036","url":null,"abstract":"<div><div>We introduce and study decompositions of graphs into so-called highly irregular graphs, as first introduced by Alavi, Chartrand, Chung, Erdős, Graham and Oellermann in the 1980s. That is, given any graph, we are interested in colouring its edges with the least number of colours possible, so that, in each colour, no vertex has two neighbours with the same degree in that colour. We provide results of different natures on this problem. We first establish connections with other notions of graph theory, including other decomposition problems, from which we notably get first bounds on the associated chromatic parameter of interest. We then study this parameter for several common classes of graphs, including graphs of bounded degree, complete bipartite graphs and complete graphs, for which we establish (sometimes close to) tight results. We also provide negative and positive algorithmic results, showing that the problem of determining our new chromatic parameter is <span>NP</span>-complete in general, but polynomial-time tractable in particular contexts. We conclude with questions and problems for further work on the topic.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115036"},"PeriodicalIF":0.9,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340466","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":"Fast exact algorithms for the SAT problem with bounded occurrences of variables","authors":"Junqiang Peng,&nbsp;Mingyu Xiao","doi":"10.1016/j.tcs.2024.115037","DOIUrl":"10.1016/j.tcs.2024.115037","url":null,"abstract":"<div><div>We present fast algorithms for the general CNF satisfiability problem (SAT) with running-time bound <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> is a function of the maximum occurrence <em>d</em> of variables (<em>d</em> can also be the average occurrence when each variable appears at least twice), and <em>n</em> is the number of variables in the input formula. Similar to SAT with bounded clause lengths, SAT with bounded occurrences of variables has also been extensively studied in the literature. Especially, the running-time bounds for small values of <em>d</em>, such as <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>d</mi><mo>=</mo><mn>4</mn></math></span>, have become bottlenecks for algorithms evaluated by the formula length <em>L</em> and other algorithms. In this paper, we show that SAT can be solved in time <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.1238</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.2628</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for <span><math><mi>d</mi><mo>=</mo><mn>4</mn></math></span>, improving the previous results <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.1279</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> and <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.2721</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> obtained by Wahlström (SAT 2005) nearly 20 years ago. For <span><math><mi>d</mi><mo>≥</mo><mn>5</mn></math></span>, we obtain a running time bound of <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.0641</mn></mrow><mrow><mi>d</mi><mi>n</mi></mrow></msup><mo>)</mo></math></span>, implying a bound of <span><math><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mn>1.0641</mn></mrow><mrow><mi>L</mi></mrow></msup><mo>)</mo></math></span> with respect to the formula length <em>L</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1029 ","pages":"Article 115037"},"PeriodicalIF":0.9,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143340470","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}