Parameterized algorithm for eternal vertex cover

In this paper we initiate the study of a “dynamic” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem introduced by Klostermeyer and Mynhardt, from the perspective of parameterized algorithms. This problem consists in placing a minimum number of guards on the vertices of...

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Veröffentlicht in:Information processing letters Jg. 110; H. 16; S. 702 - 706
Hauptverfasser: Fomin, Fedor V., Gaspers, Serge, Golovach, Petr A., Kratsch, Dieter, Saurabh, Saket
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Amsterdam Elsevier B.V 31.07.2010
Elsevier
Elsevier Sequoia S.A
Schlagworte:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Approximation
Approximations and expansions
Computer science; control theory; systems
Dynamic tests
Eternal vertex cover
Exact sciences and technology
Fixed parameter tractability
Graph algorithms
Graphs
Guards
Information processing
Information retrieval. Graph
Kernels
Mathematical analysis
Mathematics
Miscellaneous
Parameterized complexity
Placing
Running
Sciences and techniques of general use
Studies
Theoretical computing
Vertex cover
Vertex cover
Graph algorithms
Parameterized complexity
Eternal vertex cover
Fixed parameter tractability
Vertex
Edge(graph)
Computer theory
Polynomial approximation
Approximation algorithm
Complexity
Polynomial time
Response
Input
Information processing
Algorithm analysis
Graph algorithm
ISSN:0020-0190, 1872-6119
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Zusammenfassung:In this paper we initiate the study of a “dynamic” variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem introduced by Klostermeyer and Mynhardt, from the perspective of parameterized algorithms. This problem consists in placing a minimum number of guards on the vertices of a graph such that these guards can protect the graph from any sequence of attacks on its edges. In response to an attack, each guard is allowed either to stay in his vertex, or to move to a neighboring vertex. However, at least one guard has to fix the attacked edge by moving along it. The other guards may move to reconfigure and prepare for the next attack. Thus at every step the vertices occupied by guards form a vertex cover. We show that the problem admits a kernel of size 4 k ( k + 1 ) + 2 k , which shows that the problem is fixed parameter tractable when parameterized by the number of available guards k. Finally, we also provide an algorithm with running time O ( 2 O ( k 2 ) + n m ) for Eternal Vertex Cover, where n is the number of vertices and m the number of edges of the input graph. In passing we also observe that Eternal Vertex Cover is NP-hard, yet it has a polynomial time 2-approximation algorithm.
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ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2010.05.029