{"title":"有向反馈顶点集泛化的参数化算法","authors":"Alexander Göke , Dániel Marx , Matthias Mnich","doi":"10.1016/j.disopt.2022.100740","DOIUrl":null,"url":null,"abstract":"<div><p>The <span>Directed Feedback Vertex Set</span> (DFVS) problem takes as input a directed graph <span><math><mi>G</mi></math></span> and seeks a smallest vertex set <span><math><mi>S</mi></math></span> that hits all cycles in <span><math><mi>G</mi></math></span>. This is one of Karp’s 21 <span><math><mi>NP</mi></math></span>-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a <span><math><mrow><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mi>k</mi><mo>!</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>-time algorithm, where <span><math><mrow><mi>k</mi><mo>=</mo><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></math></span>.</p><p>Here we show fixed-parameter tractability of two generalizations of DFVS: </p><ul><li><span>•</span><span><p>Find a smallest vertex set <span><math><mi>S</mi></math></span> such that every strong component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> has size at most <span><math><mi>s</mi></math></span>: we give an algorithm solving this problem in time <span><math><mrow><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>k</mi><mi>s</mi><mo>+</mo><mi>k</mi><mo>+</mo><mi>s</mi><mo>)</mo></mrow><mo>!</mo><mi>⋅</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>. This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.</p></span></li><li><span>•</span><span><p>Find a smallest vertex set <span><math><mi>S</mi></math></span> such that every non-trivial strong component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> is 1-out-regular: we give an algorithm solving this problem in time <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></msup><mi>⋅</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>.</p></span></li></ul> We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.</div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Parameterized algorithms for generalizations of Directed Feedback Vertex Set\",\"authors\":\"Alexander Göke , Dániel Marx , Matthias Mnich\",\"doi\":\"10.1016/j.disopt.2022.100740\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The <span>Directed Feedback Vertex Set</span> (DFVS) problem takes as input a directed graph <span><math><mi>G</mi></math></span> and seeks a smallest vertex set <span><math><mi>S</mi></math></span> that hits all cycles in <span><math><mi>G</mi></math></span>. This is one of Karp’s 21 <span><math><mi>NP</mi></math></span>-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a <span><math><mrow><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mi>k</mi><mo>!</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>-time algorithm, where <span><math><mrow><mi>k</mi><mo>=</mo><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow></math></span>.</p><p>Here we show fixed-parameter tractability of two generalizations of DFVS: </p><ul><li><span>•</span><span><p>Find a smallest vertex set <span><math><mi>S</mi></math></span> such that every strong component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> has size at most <span><math><mi>s</mi></math></span>: we give an algorithm solving this problem in time <span><math><mrow><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup><mrow><mo>(</mo><mi>k</mi><mi>s</mi><mo>+</mo><mi>k</mi><mo>+</mo><mi>s</mi><mo>)</mo></mrow><mo>!</mo><mi>⋅</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>. This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.</p></span></li><li><span>•</span><span><p>Find a smallest vertex set <span><math><mi>S</mi></math></span> such that every non-trivial strong component of <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> is 1-out-regular: we give an algorithm solving this problem in time <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></msup><mi>⋅</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>.</p></span></li></ul> We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.</div>\",\"PeriodicalId\":50571,\"journal\":{\"name\":\"Discrete Optimization\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-11-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1572528622000457\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528622000457","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Parameterized algorithms for generalizations of Directed Feedback Vertex Set
The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph and seeks a smallest vertex set that hits all cycles in . This is one of Karp’s 21 -complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. (2008) showed its fixed-parameter tractability via a -time algorithm, where .
Here we show fixed-parameter tractability of two generalizations of DFVS:
•
Find a smallest vertex set such that every strong component of has size at most : we give an algorithm solving this problem in time . This generalizes an algorithm by Xiao (2017) for the undirected version of the problem.
•
Find a smallest vertex set such that every non-trivial strong component of is 1-out-regular: we give an algorithm solving this problem in time .
We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.
期刊介绍：
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
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