Exact and Parameterized Algorithms for Max Internal Spanning Tree

We consider the -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 with c ≤2. For graphs...

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Published in:	Algorithmica Vol. 65; no. 1; pp. 95 - 128
Main Authors:	Binkele-Raible, Daniel, Fernau, Henning, Gaspers, Serge, Liedloff, Mathieu
Format:	Journal Article
Language:	English
Published:	New York Springer-Verlag 01.01.2013 Springer Springer Verlag
Subjects:	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Data Structures and Algorithms Data Structures and Information Theory Exact sciences and technology Information retrieval. Graph Mathematics of Computing Programming languages Software Theoretical computing Theory of Computation Measure and conquer Parameterized algorithms Spanning tree problems Maximum internal spanning trees Exact exponential-time algorithms Branching Compilation Hamiltonian path Algorithmics Routing Identifier Spanning tree NP hard problem Exponential time Graph degree Dynamic programming Time complexity Algorithm analysis
ISSN:	0178-4617, 1432-0541
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Abstract	We consider the -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 with c ≤2. For graphs with bounded degree, c <2. 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 when analyzed with respect to the number of vertices. We also show that its running time is 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.
AbstractList	We consider the -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 with c ≤2. For graphs with bounded degree, c <2. 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 when analyzed with respect to the number of vertices. We also show that its running time is 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.
Author	Fernau, Henning Liedloff, Mathieu Binkele-Raible, Daniel Gaspers, Serge
Author_xml	– sequence: 1 givenname: Daniel surname: Binkele-Raible fullname: Binkele-Raible, Daniel organization: FB 4—Abteilung Informatik, Univ. Trier – sequence: 2 givenname: Henning surname: Fernau fullname: Fernau, Henning email: fernau@uni-trier.de organization: FB 4—Abteilung Informatik, Univ. Trier – sequence: 3 givenname: Serge surname: Gaspers fullname: Gaspers, Serge organization: Institute of Information Systems (184/3), Vienna Univ. of Technology – sequence: 4 givenname: Mathieu surname: Liedloff fullname: Liedloff, Mathieu organization: LIFO, Univ. d’Orléans
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Keywords	Measure and conquer Parameterized algorithms Spanning tree problems Maximum internal spanning trees Exact exponential-time algorithms Branching Compilation Hamiltonian path Algorithmics Routing Identifier Spanning tree NP hard problem Exponential time Graph degree Dynamic programming Time complexity Algorithm analysis
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Snippet	We consider the -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Data Structures and Algorithms Data Structures and Information Theory Exact sciences and technology Information retrieval. Graph Mathematics of Computing Programming languages Software Theoretical computing Theory of Computation
Title	Exact and Parameterized Algorithms for Max Internal Spanning Tree
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