On the approximability and exact algorithms for vector domination and related problems in graphs

Authors:
Ferdinando Cicalese;Martin Milanič;Ugo Vaccaro
Affiliations:
Dipartimento di Informatica, University of Salerno, 84084 Fisciano (SA), Italy;University of Primorska, UP IAM, Muzejski trg 2, SI6000 Koper, Slovenia and University of Primorska, UP FAMNIT, Glagoljaška 8, SI6000 Koper, Slovenia;Dipartimento di Informatica, University of Salerno, 84084 Fisciano (SA), Italy
Venue:
Discrete Applied Mathematics
Year:
2013

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Abstract

We consider two graph optimization problems called vector domination and total vector domination. In vector domination one seeks a small subset S of vertices of a graph such that any vertex outside S has a prescribed number of neighbors in S. In total vector domination, the requirement is extended to all vertices of the graph. We prove that these problems (and several variants thereof) cannot be approximated to within a factor of clnn, where c is a suitable constant and n is the number of the vertices, unless P=NP. We also show that two natural greedy strategies have approximation factors ln@D+O(1), where @D is the maximum degree of the input graph. We also provide exact polynomial time algorithms for several classes of graphs. Our results extend, improve, and unify several results previously known in the literature.