On positive-influence target-domination

Consider a graph G = (V, E) and a vertex subset A subset of V. A vertex upsilon is positive-influence dominated by A if either upsilon is in A or at least half the number of neighbors of upsilon belong to A. For a target vertex subset S subset of V, a vertex subset A is a positive-influence target-dominating set for target set S if every vertex in S is positive-influence dominated by A. Given a graph G and a target vertex subset S, the positive-influence target-dominating set (PITD) problem is to find the minimum positive-influence dominating set for target S. In this paper, we show two results: (1) The PITD problem has a polynomial-time (1 + log inverted right perpendicular 3/2 Delta inverted right perpendicular)-approximation in general graphs where Delta is the maximum vertex-degree of the input graph. (2) For target set S with broken vertical bar S broken vertical bar = Omega(broken vertical bar V broken vertical bar), the PITD problem has a polynomial-time O(1)-approximation in power-law graphs.