Minimum Monopoly in Regular and Tree Graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S ⊆ V, such that, for each u ∈ V, ∣N[u]∩ S∣≤ ∣N[u]/2 in a given graph G = (V, E). We show that this optimization problem does not have a polynomial time approximation scheme for k-regular graphs (k ≥ 5), unless P = NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for fe-regular graphs to minimum monoposyly problem for 2fe-regular graphs and to minimum monopoly problem for (2k -1)-regular graphs, where k ≥ 3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.