Approximation algorithm for the minimum partial connected Roman dominating set problem

Abstract

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