Approximation algorithm for the minimum partial connected Roman dominating set problem

Given a graph G = (V, E) and a function r : V -> {0, 1, 2}, a node v is an element of V is said to be Roman dominated if r(v) = 1 or there exists a node u is an element of N-G[v] such that r(u) = 2, where N-G[v] is the closed neighbor set of v in G. For i is an element of {0, 1, 2}, denote V-r(i) as the set of nodes with value i under function r. The cost of r is defined to be c(r) = vertical bar V-r(1)vertical bar + 2 vertical bar V-r(2)vertical bar. Given a positive integer Q <= vertical bar V vertical bar, the minimum partial connected Roman dominating set (MinPCRDS) problem is to compute aminimum cost function r such that at least Q nodes in G are Roman dominated and the subgraph of G induced by V-r(1) boolean OR V-r(2) is connected. In this paper, we give a (3 ln vertical bar V vertical bar + 9)-approximation algorithm for the MinPCRDS problem.