Approximation algorithm for connected Roman k-dominating set

Abstract

Let $k(k\ge 2)$ be a positive integer, and let G be a simple graph with the vertex set $V\left(G\right)$. A Roman k-dominating function (Rk-DF) on G is a function $f_{kR}:V\left(G\right)\to \{0,1,2\}$ such that every vertex u with $f_{kR}\left(u\right)=0$ is adjacent to at least k vertices $v_1,v_2,\dots ,v_k$ with $f_{kR}\left(v_i\right)=2$ for $i=1,2,\dots ,k$. The minimum Roman k-dominating set problem aims to compute a Roman k-dominating function $f_{kR}$ that minimizes the total weight ${\sum }_{v\in V}{f}_{kR}\left(v\right)$. The minimum Connected Roman k-dominating set problem (MinCRkDS) seeks to find a minimum-weight Roman k-dominating function $f_{kR}$ such that the subgraph of G induced by $D_{kR}=\{v\in V|f_{kR}\left(v\right)=1\text{ or }{f}_{kR}\left(v\right)=2\}$ is connected. As far as we know, this paper is the first paper to solve MinCRkDS in general graphs. We present a greedy algorithm for MinCRkDS with an approximation ratio $(1+\epsilon )(2+\ln (k+1+2\Delta \left)\right)$ for any $\epsilon >0$, where Δ is the maximum degree of the graph.