We consider approximability of two natural variants of classical dominating set problem, namely minimum majority monopoly and minimum signed domination. In the minimum majority monopoly problem, the objective is to find a smallest size subset X ⊆ V in a given graph G = ( V , E ) such that | N [ v ] ∩ X | ⩾ 1 2 | N [ v ] | , for at least half of the vertices in V. On the other hand, given a graph G = ( V , E ) , in the signed domination problem one needs to find a function f : V → { − 1 , 1 } such that f ( N [ v ] ) ⩾ 1 , for all v ∈ V , and the cost f ( V ) = ∑ v ∈ V f ( v ) is minimized. We show that minimum majority monopoly and minimum signed domination cannot be approximated within a factor of ( 1 2 − ϵ ) ln n and ( 1 3 − ϵ ) ln n , respectively, for any ϵ > 0 , unless NP ⊆ Dtime ( n O ( log log n ) ) . We also prove that, if Δ is the maximum degree of a vertex in the graph, then both problems cannot be approximated within a factor of ln Δ − D ln ln Δ , for some constant D, unless P = NP . On the positive side, we give ln ( Δ + 1 ) -factor approximation algorithm for minimum majority monopoly problem for general graphs. We show that minimum majority monopoly problem is APX -complete for graphs with degree at most 3 and at least 2 and minimum signed domination problem is APX -complete, for 3-regular graphs. For 3-regular graphs, these two problems are approximable within a factor of 4 3 (asymptotically) and 1.6, respectively.