Nearly tight approximation algorithm for (connected) Roman dominating set

Roman dominating function of graph G is a function r: V(G)→{0, 1, 2} satisfying that every vertex v with r(v)=0 is adjacent to at least one vertex u with r(u)=2. The minimum Roman dominating set problem (MinRDS) is to compute a Roman dominating function r that minimizes the weight ∑ _v∈ Vr(v). The minimum connected Roman dominating set problem (MinCRDS) is to find a minimum weight Roman dominating function r_c such that the subgraph of G induced by D_R={v ∈ V | r_c(v)=1 or r_c(v)=2} is connected. In this paper, we present a greedy algorithm for MinRDS with a guaranteed performance ratio at most H(δ _max+1), where H(· ) is the Harmonic number and δ _max is the maximum degree of the graph. For any ε >0, we show that there exists a greedy algorithm for MinCRDS with approximation ratio at most (1+ε )lnδ _max+O(1). The challenge for the analysis of the MinCRDS algorithm lies in the fact that the potential function is not only non-submodular but also non-monotone.