Theoretical Computer Science最新文献_第5页

Super spanning connectivity of the generalized hypercube network

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-20 DOI: 10.1016/j.tcs.2024.115038

Xiaoqian Wang, Eminjan Sabir

引用次数: 0

Highly irregular graph decompositions

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-20 DOI: 10.1016/j.tcs.2024.115036

Julien Bensmail , Malory Marin , Leandro Montero , Alexandre Talon

{"title":"Highly irregular graph decompositions","authors":"Julien Bensmail , Malory Marin , Leandro Montero , 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}

引用次数: 0

Fast exact algorithms for the SAT problem with bounded occurrences of variables

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-19 DOI: 10.1016/j.tcs.2024.115037

Junqiang Peng, Mingyu Xiao

{"title":"Fast exact algorithms for the SAT problem with bounded occurrences of variables","authors":"Junqiang Peng, 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}

引用次数: 0

Approximation hardness of domination problems on generalized convex graphs

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-17 DOI: 10.1016/j.tcs.2024.115035

Po Yuan Wang, Naoki Kitamura, Taisuke Izumi, Toshimitsu Masuzawa

{"title":"Approximation hardness of domination problems on generalized convex graphs","authors":"Po Yuan Wang, Naoki Kitamura, Taisuke Izumi, Toshimitsu Masuzawa","doi":"10.1016/j.tcs.2024.115035","DOIUrl":"10.1016/j.tcs.2024.115035","url":null,"abstract":"<div><div>The domination problem and its variants in bipartite graphs are computationally challenging, known to be <span><math><mi>NP</mi></math></span>-complete and hard to approximate. However, for convex bipartite graphs, it becomes polynomial-time solvable, raising questions about the boundaries of tractability and intractability, as well as approximability and inapproximability, within bipartite graph subclasses. This study examines the approximation hardness of the domination problem for generalized convex graphs, a subclass of bipartite graphs that extends convex bipartite graphs. We explore the approximation hardness for various domination problem variants, including total, connected, paired, and independent domination. Previous research has highlighted the critical role of the <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>Δ</mi><mo>)</mo></math></span>-tree convex graph parameters in determining computational complexity, demonstrating polynomial-time solvability when both <em>t</em> and Δ are bounded. Extending these findings, our research establishes that unbounded <em>t</em> or Δ results in <span><math><mi>APX</mi></math></span>-hardness for all examined domination variants. Notably, this result encompasses star and comb convex bipartite graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1028 ","pages":"Article 115035"},"PeriodicalIF":0.9,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143169001","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}

引用次数: 0

Termination of rewriting on reversible Boolean circuits as a free 3-category problem

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-17 DOI: 10.1016/j.tcs.2024.115031

Adriano Barile, Stefano Berardi, Luca Roversi

引用次数: 0

Improved FPT approximation scheme and approximate kernel for biclique-free max k-weight SAT: Greedy strikes back

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-17 DOI: 10.1016/j.tcs.2024.115033

Pasin Manurangsi

{"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}

引用次数: 0

Parameterized algorithms for minimum sum vertex cover

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-17 DOI: 10.1016/j.tcs.2024.115032

Shubhada Aute , Fahad Panolan

{"title":"Parameterized algorithms for minimum sum vertex cover","authors":"Shubhada Aute , 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}

引用次数: 0

Faster parameterized algorithm for r-pseudoforest deletion

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-16 DOI: 10.1016/j.tcs.2024.115034

Dekel Tsur

引用次数: 0

Parameterizing path partitions

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-14 DOI: 10.1016/j.tcs.2024.115029

Henning Fernau , Florent Foucaud , Kevin Mann , Utkarsh Padariya , Rajath Rao K.N.

{"title":"Parameterizing path partitions","authors":"Henning Fernau , Florent Foucaud , Kevin Mann , Utkarsh Padariya , 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}

引用次数: 0

Payment scheduling in the Interval Debt Model

IF 0.9 4区计算机科学

Theoretical Computer Science Pub Date : 2024-12-13 DOI: 10.1016/j.tcs.2024.115028

Tom Friedetzky, David C. Kutner, George B. Mertzios , Iain A. Stewart, Amitabh Trehan

{"title":"Payment scheduling in the Interval Debt Model","authors":"Tom Friedetzky, David C. Kutner, George B. Mertzios , Iain A. Stewart, 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}

引用次数: 0