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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Bibliographic Details
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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Summary: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.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-011-9575-5