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 k-regular graphs to minimum monopoly problem for 2 k -regular graphs and to minimum monopoly problem for ( 2 k - 1 ) -regular graphs, where k ⩾ 3 . We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.