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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Amsterdam
Elsevier B.V
31.07.2010
Elsevier Elsevier Sequoia S.A |
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| Abstract | 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. |
|---|---|
| AbstractList | In this paper the authors 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. The authors show that the problem admits a kernel of size 4...(k+1)+2k, which shows that the problem is fixed parameter tractable when parameterized by the number of available guards k. Finally, they also provide an algorithm with running time ... for Eternal Vertex Cover, where n is the number of vertices and m the number of edges of the input graph. In passing they also observe that Eternal Vertex Cover is NP-hard, yet it has a polynomial time 2-approximation algorithm. (ProQuest: ... denotes formulae/symbols omitted.) 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 super()kk+1)+2k, 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 [MathML equation] 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. 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. |
| Author | Fomin, Fedor V. Golovach, Petr A. Gaspers, Serge Kratsch, Dieter Saurabh, Saket |
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| Cites_doi | 10.1109/SFCS.1980.12 10.1007/s00453-007-9148-9 10.1016/j.jcss.2009.04.001 10.1007/s00224-007-9089-3 10.1007/s00224-007-1309-3 |
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| Keywords | 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 |
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Vertex Cover problem, the
Eternal Vertex Cover problem introduced by Klostermeyer... In this paper the authors initiate the study of a "dynamic" variant of the classical Vertex Cover problem, the Eternal Vertex Cover problem introduced by... 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... |
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| SubjectTerms | 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 |
| Title | Parameterized algorithm for eternal vertex cover |
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