Exact and Parameterized Algorithms for Max Internal Spanning Tree
Identifieur interne : 001A70 ( Istex/Corpus ); précédent : 001A69; suivant : 001A71Exact and Parameterized Algorithms for Max Internal Spanning Tree
Auteurs : Henning Fernau ; Serge Gaspers ; Daniel RaibleSource :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 2010.
Abstract
Abstract: We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O *(c n ) (c ≤ 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8669 n ) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364 k n O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.
Url:
DOI: 10.1007/978-3-642-11409-0_9
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Trier, FB 4—Abteilung Informatik, D-54286, Trier, Germany</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: raible@uni-trier.de</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="s">Lecture Notes in Computer Science</title><imprint><date>2010</date></imprint><idno type="ISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="ISSN">0302-9743</idno></series><idno type="istex">E33D47F68C87766E4ABEE948C04444BA98EEFFD5</idno><idno type="DOI">10.1007/978-3-642-11409-0_9</idno><idno type="ChapterID">9</idno><idno type="ChapterID">Chap9</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0302-9743</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O *(c n ) (c ≤ 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8669 n ) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364 k n O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>Henning Fernau</name><affiliations><json:string>Univ. Trier, FB 4—Abteilung Informatik, D-54286, Trier, Germany</json:string><json:string>E-mail: fernau@uni-trier.de</json:string></affiliations></json:item><json:item><name>Serge Gaspers</name><affiliations><json:string>LIRMM – Univ. of Montpellier 2, CNRS, 34392, Montpellier, France</json:string><json:string>E-mail: gaspers@lirmm.fr</json:string></affiliations></json:item><json:item><name>Daniel Raible</name><affiliations><json:string>Univ. Trier, FB 4—Abteilung Informatik, D-54286, Trier, Germany</json:string><json:string>E-mail: raible@uni-trier.de</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>OriginalPaper</json:string></originalGenre><abstract>Abstract: We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O *(c n ) (c ≤ 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8669 n ) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364 k n O(1) when the goal is to find a spanning tree with at least k internal vertices. 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Kleinberg</name><affiliations><json:string>Cornell University, Ithaca, NY, USA</json:string></affiliations></json:item><json:item><name>Friedemann Mattern</name><affiliations><json:string>ETH Zurich, Zurich, Switzerland</json:string></affiliations></json:item><json:item><name>John C. Mitchell</name><affiliations><json:string>Stanford University, Stanford, CA, USA</json:string></affiliations></json:item><json:item><name>Moni Naor</name><affiliations><json:string>Weizmann Institute of Science, Rehovot, Israel</json:string></affiliations></json:item><json:item><name>Oscar Nierstrasz</name><affiliations><json:string>University of Bern, Bern, Switzerland</json:string></affiliations></json:item><json:item><name>C. Pandu Rangan</name><affiliations><json:string>Indian Institute of Technology, Madras, India</json:string></affiliations></json:item><json:item><name>Bernhard Steffen</name><affiliations><json:string>University of Dortmund, Dortmund, Germany</json:string></affiliations></json:item><json:item><name>Madhu Sudan</name><affiliations><json:string>Massachusetts Institute of Technology, MA, USA</json:string></affiliations></json:item><json:item><name>Demetri Terzopoulos</name><affiliations><json:string>University of California, Los Angeles, CA, USA</json:string></affiliations></json:item><json:item><name>Doug Tygar</name><affiliations><json:string>University of California, Berkeley, CA, USA</json:string></affiliations></json:item><json:item><name>Moshe Y. Vardi</name><affiliations><json:string>Rice University, Houston, TX, USA</json:string></affiliations></json:item><json:item><name>Gerhard Weikum</name><affiliations><json:string>Max-Planck Institute of Computer Science, Saarbrücken, Germany</json:string></affiliations></json:item></editor><issn><json:string>0302-9743</json:string></issn><language><json:string>unknown</json:string></language><eissn><json:string>1611-3349</json:string></eissn><title>Lecture Notes in Computer Science</title><copyrightDate>2010</copyrightDate></serie><host><editor><json:item><name>Christophe Paul</name><affiliations><json:string>CNRS - LIRMM, 161 rue Ada, 34392, Montpellier Cedex 5, France</json:string><json:string>E-mail: christophe.paul@lirmm.fr</json:string></affiliations></json:item><json:item><name>Michel Habib</name><affiliations><json:string>LIAFA, case 7014, Université Paris Diderot, 75205, Paris Cedex 13, France</json:string><json:string>E-mail: habib@liafa.jussieu.fr</json:string></affiliations></json:item></editor><subject><json:item><value>Computer Science</value></json:item><json:item><value>Computer Science</value></json:item><json:item><value>Discrete Mathematics in Computer Science</value></json:item><json:item><value>Geometry</value></json:item><json:item><value>Algorithms</value></json:item><json:item><value>Symbolic and Algebraic Manipulation</value></json:item><json:item><value>Algorithm Analysis and Problem Complexity</value></json:item><json:item><value>Data Structures</value></json:item></subject><isbn><json:string>978-3-642-11408-3</json:string></isbn><language><json:string>unknown</json:string></language><eissn><json:string>1611-3349</json:string></eissn><title>Graph-Theoretic Concepts in Computer Science</title><bookId><json:string>978-3-642-11409-0</json:string></bookId><volume>5911</volume><pages><last>111</last><first>100</first></pages><issn><json:string>0302-9743</json:string></issn><genre><json:string>book-series</json:string></genre><eisbn><json:string>978-3-642-11409-0</json:string></eisbn><copyrightDate>2010</copyrightDate><doi><json:string>10.1007/978-3-642-11409-0</json:string></doi></host><publicationDate>2010</publicationDate><copyrightDate>2010</copyrightDate><doi><json:string>10.1007/978-3-642-11409-0_9</json:string></doi><id>E33D47F68C87766E4ABEE948C04444BA98EEFFD5</id><score>0.434334</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/document/E33D47F68C87766E4ABEE948C04444BA98EEFFD5/fulltext/pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/document/E33D47F68C87766E4ABEE948C04444BA98EEFFD5/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/E33D47F68C87766E4ABEE948C04444BA98EEFFD5/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Exact and Parameterized Algorithms for Max Internal Spanning Tree</title><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt><respStmt><resp>Références bibliographiques récupérées via GROBID</resp><name resp="ISTEX-API">ISTEX-API (INIST-CNRS)</name></respStmt></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>Springer Berlin Heidelberg</publisher><pubPlace>Berlin, Heidelberg</pubPlace><availability><p>Springer-Verlag Berlin Heidelberg, 2010</p></availability><date>2010</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Exact and Parameterized Algorithms for Max Internal Spanning Tree</title><author xml:id="author-1"><persName><forename type="first">Henning</forename><surname>Fernau</surname></persName><email>fernau@uni-trier.de</email><affiliation>Univ. Trier, FB 4—Abteilung Informatik, D-54286, Trier, Germany</affiliation></author><author xml:id="author-2"><persName><forename type="first">Serge</forename><surname>Gaspers</surname></persName><email>gaspers@lirmm.fr</email><affiliation>LIRMM – Univ. of Montpellier 2, CNRS, 34392, Montpellier, France</affiliation></author><author xml:id="author-3"><persName><forename type="first">Daniel</forename><surname>Raible</surname></persName><email>raible@uni-trier.de</email><affiliation>Univ. Trier, FB 4—Abteilung Informatik, D-54286, Trier, Germany</affiliation></author></analytic><monogr><title level="m">Graph-Theoretic Concepts in Computer Science</title><title level="m" type="sub">35th International Workshop, WG 2009, Montpellier, France, June 24-26, 2009. Revised Papers</title><idno type="pISBN">978-3-642-11408-3</idno><idno type="eISBN">978-3-642-11409-0</idno><idno type="pISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="DOI">10.1007/978-3-642-11409-0</idno><idno type="book-ID">978-3-642-11409-0</idno><idno type="book-title-ID">194091</idno><idno type="book-sequence-number">5911</idno><idno type="book-volume-number">5911</idno><idno type="book-chapter-count">30</idno><editor><persName><forename type="first">Christophe</forename><surname>Paul</surname></persName><email>christophe.paul@lirmm.fr</email><affiliation>CNRS - LIRMM, 161 rue Ada, 34392, Montpellier Cedex 5, France</affiliation></editor><editor><persName><forename type="first">Michel</forename><surname>Habib</surname></persName><email>habib@liafa.jussieu.fr</email><affiliation>LIAFA, case 7014, Université Paris Diderot, 75205, Paris Cedex 13, France</affiliation></editor><imprint><publisher>Springer Berlin Heidelberg</publisher><pubPlace>Berlin, Heidelberg</pubPlace><date type="published" when="2010"></date><biblScope unit="volume">5911</biblScope><biblScope unit="page" from="100">100</biblScope><biblScope unit="page" to="111">111</biblScope></imprint></monogr><series><title level="s">Lecture Notes in Computer Science</title><editor><persName><forename type="first">David</forename><surname>Hutchison</surname></persName><affiliation>Lancaster University, Lancaster, UK</affiliation></editor><editor><persName><forename type="first">Takeo</forename><surname>Kanade</surname></persName><affiliation>Carnegie Mellon University, Pittsburgh, PA, USA</affiliation></editor><editor><persName><forename type="first">Josef</forename><surname>Kittler</surname></persName><affiliation>University of Surrey, Guildford, UK</affiliation></editor><editor><persName><forename type="first">Jon</forename><forename type="first">M.</forename><surname>Kleinberg</surname></persName><affiliation>Cornell University, Ithaca, NY, USA</affiliation></editor><editor><persName><forename type="first">Friedemann</forename><surname>Mattern</surname></persName><affiliation>ETH Zurich, Zurich, Switzerland</affiliation></editor><editor><persName><forename type="first">John</forename><forename type="first">C.</forename><surname>Mitchell</surname></persName><affiliation>Stanford University, Stanford, CA, USA</affiliation></editor><editor><persName><forename type="first">Moni</forename><surname>Naor</surname></persName><affiliation>Weizmann Institute of Science, Rehovot, Israel</affiliation></editor><editor><persName><forename type="first">Oscar</forename><surname>Nierstrasz</surname></persName><affiliation>University of Bern, Bern, Switzerland</affiliation></editor><editor><persName><forename type="first">C.</forename><surname>Pandu Rangan</surname></persName><affiliation>Indian Institute of Technology, Madras, India</affiliation></editor><editor><persName><forename type="first">Bernhard</forename><surname>Steffen</surname></persName><affiliation>University of Dortmund, Dortmund, Germany</affiliation></editor><editor><persName><forename type="first">Madhu</forename><surname>Sudan</surname></persName><affiliation>Massachusetts Institute of Technology, MA, USA</affiliation></editor><editor><persName><forename type="first">Demetri</forename><surname>Terzopoulos</surname></persName><affiliation>University of California, Los Angeles, CA, USA</affiliation></editor><editor><persName><forename type="first">Doug</forename><surname>Tygar</surname></persName><affiliation>University of California, Berkeley, CA, USA</affiliation></editor><editor><persName><forename type="first">Moshe</forename><forename type="first">Y.</forename><surname>Vardi</surname></persName><affiliation>Rice University, Houston, TX, USA</affiliation></editor><editor><persName><forename type="first">Gerhard</forename><surname>Weikum</surname></persName><affiliation>Max-Planck Institute of Computer Science, Saarbrücken, Germany</affiliation></editor><biblScope><date>2010</date></biblScope><idno type="pISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="series-Id">558</idno></series><idno type="istex">E33D47F68C87766E4ABEE948C04444BA98EEFFD5</idno><idno type="DOI">10.1007/978-3-642-11409-0_9</idno><idno type="ChapterID">9</idno><idno type="ChapterID">Chap9</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2010</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>Abstract: We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O *(c n ) (c ≤ 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8669 n ) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364 k n O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.</p></abstract><textClass><keywords scheme="Book-Subject-Collection"><list><label>SUCO11645</label><item><term>Computer Science</term></item></list></keywords></textClass><textClass><keywords scheme="Book-Subject-Group"><list><label>I</label><label>I17028</label><label>M21006</label><label>M14018</label><label>I17052</label><label>I16021</label><label>I15017</label><item><term>Computer Science</term></item><item><term>Discrete Mathematics in Computer Science</term></item><item><term>Geometry</term></item><item><term>Algorithms</term></item><item><term>Symbolic and Algebraic Manipulation</term></item><item><term>Algorithm Analysis and Problem Complexity</term></item><item><term>Data Structures</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="2010">Published</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2016-11-22">References added</change><change xml:id="refBibs-istex" who="#ISTEX-API" when="2017-01-20">References added</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/E33D47F68C87766E4ABEE948C04444BA98EEFFD5/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Springer, Publisher found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Springer Berlin Heidelberg</PublisherName><PublisherLocation>Berlin, Heidelberg</PublisherLocation></PublisherInfo><Series><SeriesInfo SeriesType="Series" 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DisplayOrder="Western"><GivenName>Madhu</GivenName><FamilyName>Sudan</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff12"><EditorName DisplayOrder="Western"><GivenName>Demetri</GivenName><FamilyName>Terzopoulos</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff13"><EditorName DisplayOrder="Western"><GivenName>Doug</GivenName><FamilyName>Tygar</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff14"><EditorName DisplayOrder="Western"><GivenName>Moshe</GivenName><GivenName>Y.</GivenName><FamilyName>Vardi</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff15"><EditorName DisplayOrder="Western"><GivenName>Gerhard</GivenName><FamilyName>Weikum</FamilyName></EditorName></Editor><Affiliation ID="Aff1"><OrgName>Lancaster University</OrgName><OrgAddress><City>Lancaster</City><Country>UK</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>Carnegie Mellon University</OrgName><OrgAddress><City>Pittsburgh</City><State>PA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgName>University of Surrey</OrgName><OrgAddress><City>Guildford</City><Country>UK</Country></OrgAddress></Affiliation><Affiliation ID="Aff4"><OrgName>Cornell University</OrgName><OrgAddress><City>Ithaca</City><State>NY</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff5"><OrgName>ETH Zurich</OrgName><OrgAddress><City>Zurich</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff6"><OrgName>Stanford University</OrgName><OrgAddress><City>Stanford</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff7"><OrgName>Weizmann Institute of Science</OrgName><OrgAddress><City>Rehovot</City><Country>Israel</Country></OrgAddress></Affiliation><Affiliation ID="Aff8"><OrgName>University of Bern</OrgName><OrgAddress><City>Bern</City><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation ID="Aff9"><OrgName>Indian Institute of Technology</OrgName><OrgAddress><City>Madras</City><Country>India</Country></OrgAddress></Affiliation><Affiliation ID="Aff10"><OrgName>University of Dortmund</OrgName><OrgAddress><City>Dortmund</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff11"><OrgName>Massachusetts Institute of Technology</OrgName><OrgAddress><State>MA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff12"><OrgName>University of California</OrgName><OrgAddress><City>Los Angeles</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff13"><OrgName>University of California</OrgName><OrgAddress><City>Berkeley</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff14"><OrgName>Rice University</OrgName><OrgAddress><City>Houston</City><State>TX</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff15"><OrgName>Max-Planck Institute of Computer Science</OrgName><OrgAddress><City>Saarbrücken</City><Country>Germany</Country></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Proceedings" ContainsESM="No" Language="En" MediaType="eBook" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><BookID>978-3-642-11409-0</BookID><BookTitle>Graph-Theoretic Concepts in Computer Science</BookTitle><BookSubTitle>35th International Workshop, WG 2009, Montpellier, France, June 24-26, 2009. Revised Papers</BookSubTitle><BookVolumeNumber>5911</BookVolumeNumber><BookSequenceNumber>5911</BookSequenceNumber><BookDOI>10.1007/978-3-642-11409-0</BookDOI><BookTitleID>194091</BookTitleID><BookPrintISBN>978-3-642-11408-3</BookPrintISBN><BookElectronicISBN>978-3-642-11409-0</BookElectronicISBN><BookChapterCount>30</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2010</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I17028" Priority="1" Type="Secondary">Discrete Mathematics in Computer Science</BookSubject><BookSubject Code="M21006" Priority="2" Type="Secondary">Geometry</BookSubject><BookSubject Code="M14018" Priority="3" Type="Secondary">Algorithms</BookSubject><BookSubject Code="I17052" Priority="4" Type="Secondary">Symbolic and Algebraic Manipulation</BookSubject><BookSubject Code="I16021" Priority="5" Type="Secondary">Algorithm Analysis and Problem Complexity</BookSubject><BookSubject Code="I15017" Priority="6" Type="Secondary">Data Structures</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>558</SeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Christophe</GivenName><FamilyName>Paul</FamilyName></EditorName><Contact><Email>christophe.paul@lirmm.fr</Email></Contact></Editor><Editor AffiliationIDS="Aff17"><EditorName DisplayOrder="Western"><GivenName>Michel</GivenName><FamilyName>Habib</FamilyName></EditorName><Contact><Email>habib@liafa.jussieu.fr</Email></Contact></Editor><Affiliation ID="Aff16"><OrgName>CNRS - LIRMM</OrgName><OrgAddress><Street>161 rue Ada</Street><Postcode>34392</Postcode><City>Montpellier Cedex 5</City><Country>France</Country></OrgAddress></Affiliation><Affiliation ID="Aff17"><OrgName>LIAFA, case 7014, Université Paris Diderot</OrgName><OrgAddress><Postcode>75205</Postcode><City>Paris Cedex 13</City><Country>France</Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Chapter ID="Chap9" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingStyle="ContentOnly" TocLevels="0"><ChapterID>9</ChapterID><ChapterDOI>10.1007/978-3-642-11409-0_9</ChapterDOI><ChapterSequenceNumber>9</ChapterSequenceNumber><ChapterTitle Language="En">Exact and Parameterized Algorithms for <Emphasis Type="SmallCaps">Max Internal Spanning Tree</Emphasis></ChapterTitle><ChapterFirstPage>100</ChapterFirstPage><ChapterLastPage>111</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2010</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><BookID>978-3-642-11409-0</BookID><BookTitle>Graph-Theoretic Concepts in Computer Science</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff18"><AuthorName DisplayOrder="Western"><GivenName>Henning</GivenName><FamilyName>Fernau</FamilyName></AuthorName><Contact><Email>fernau@uni-trier.de</Email></Contact></Author><Author AffiliationIDS="Aff19"><AuthorName DisplayOrder="Western"><GivenName>Serge</GivenName><FamilyName>Gaspers</FamilyName></AuthorName><Contact><Email>gaspers@lirmm.fr</Email></Contact></Author><Author AffiliationIDS="Aff18"><AuthorName DisplayOrder="Western"><GivenName>Daniel</GivenName><FamilyName>Raible</FamilyName></AuthorName><Contact><Email>raible@uni-trier.de</Email></Contact></Author><Affiliation ID="Aff18"><OrgName>Univ. Trier, FB 4—Abteilung Informatik</OrgName><OrgAddress><Postcode>D-54286</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff19"><OrgName>LIRMM – Univ. of Montpellier 2, CNRS</OrgName><OrgAddress><Postcode>34392</Postcode><City>Montpellier</City><Country>France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>We consider the <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject FileRef="978-3-642-11409-0_9_Chapter_TeX2GIFIEq1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{NP}$</EquationSource></InlineEquation>-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous <Emphasis Type="SmallCaps">Hamiltonian Path</Emphasis> problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form <Emphasis Type="Italic">O</Emphasis><Superscript>*</Superscript>(<Emphasis Type="Italic">c</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>) (<Emphasis Type="Italic">c</Emphasis> ≤ 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of <Emphasis Type="Italic">O</Emphasis>(1.8669<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript><Emphasis Type="Italic">n</Emphasis><Superscript><Emphasis Type="Italic">O</Emphasis>(1)</Superscript> when the goal is to find a spanning tree with at least <Emphasis Type="Italic">k</Emphasis> internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>This work was partially Supported by a PPP grant between DAAD (Germany) and NFR (Norway).</SimplePara></ArticleNote></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Exact and Parameterized Algorithms for Max Internal Spanning Tree</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Exact and Parameterized Algorithms for Max Internal Spanning Tree</title></titleInfo><name type="personal"><namePart type="given">Henning</namePart><namePart type="family">Fernau</namePart><affiliation>Univ. Trier, FB 4—Abteilung Informatik, D-54286, Trier, Germany</affiliation><affiliation>E-mail: fernau@uni-trier.de</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Serge</namePart><namePart type="family">Gaspers</namePart><affiliation>LIRMM – Univ. of Montpellier 2, CNRS, 34392, Montpellier, France</affiliation><affiliation>E-mail: gaspers@lirmm.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Daniel</namePart><namePart type="family">Raible</namePart><affiliation>Univ. Trier, FB 4—Abteilung Informatik, D-54286, Trier, Germany</affiliation><affiliation>E-mail: raible@uni-trier.de</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="conference" displayLabel="OriginalPaper"></genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2010</dateIssued><copyrightDate encoding="w3cdtf">2010</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O *(c n ) (c ≤ 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O(1.8669 n ) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364 k n O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.</abstract><relatedItem type="host"><titleInfo><title>Graph-Theoretic Concepts in Computer Science</title><subTitle>35th International Workshop, WG 2009, Montpellier, France, June 24-26, 2009. 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type="family">Mitchell</namePart><affiliation>Stanford University, Stanford, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Moni</namePart><namePart type="family">Naor</namePart><affiliation>Weizmann Institute of Science, Rehovot, Israel</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Oscar</namePart><namePart type="family">Nierstrasz</namePart><affiliation>University of Bern, Bern, Switzerland</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">C.</namePart><namePart type="family">Pandu Rangan</namePart><affiliation>Indian Institute of Technology, Madras, India</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Bernhard</namePart><namePart type="family">Steffen</namePart><affiliation>University of Dortmund, Dortmund, Germany</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Madhu</namePart><namePart type="family">Sudan</namePart><affiliation>Massachusetts Institute of Technology, MA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Demetri</namePart><namePart type="family">Terzopoulos</namePart><affiliation>University of California, Los Angeles, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Doug</namePart><namePart type="family">Tygar</namePart><affiliation>University of California, Berkeley, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Moshe</namePart><namePart type="given">Y.</namePart><namePart type="family">Vardi</namePart><affiliation>Rice University, Houston, TX, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Gerhard</namePart><namePart type="family">Weikum</namePart><affiliation>Max-Planck Institute of Computer Science, Saarbrücken, Germany</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><copyrightDate encoding="w3cdtf">2010</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="SeriesID">558</identifier><recordInfo><recordOrigin>Springer-Verlag Berlin Heidelberg, 2010</recordOrigin></recordInfo></relatedItem><identifier type="istex">E33D47F68C87766E4ABEE948C04444BA98EEFFD5</identifier><identifier type="DOI">10.1007/978-3-642-11409-0_9</identifier><identifier type="ChapterID">9</identifier><identifier type="ChapterID">Chap9</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag Berlin 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