We consider new variants of the vertex/edge domination problems on graphs. A vertex is said to dominate itself and its all adjacent vertices, and similarly an edge is said to dominate itself and its all adjacent edges. Given an input graph G = (V, E) and an integer k, the k-Vertex (k-Edge) Maximum Domination (k-MaxVD and k-MaxED, respectively) is to find a subset DV ⊆ V of vertices (resp., DE ⊆ E of edges) with size at most k that maximizes the cardinality of dominated vertices (resp., edges). In this paper, we first show that a simple greedy strategy achieves an approximation ratio of (1 − 1/e) for both k-MaxVD and k-MaxED. Then, we show that this approximation ratio is the best possible for k-MaxVD unless P = NP. We also prove that, for any constant ε > 0, there is no polynomial time 1303/1304+ε approximation algorithm for k-MaxED unless P = NP. However, if k is not larger than the size of the minimum maximal matching, k-MaxED is 3/4-approximable in polynomial time.
Cite as: Miyano, E. and Ono, H. (2011). Maximum Domination Problem. In Proc. Computing: The Australasian Theory Symposium (CATS 2011) Perth, Australia. CRPIT, 119. Alex Potanin and Taso Viglas Eds., ACS. 55-62
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