Nearly tight approximation algorithm for (connected) Roman dominating set

Abstract

A Roman dominating function of graph G is a function $r: V(G)\rightarrow \{0, 1, 2\}$ 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 $\sum _{v\in V}r(v).$ 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 $D_{R}=\{v \in V \mid r_{c}(v)=1 ~\mathrm{{or}}~ r_{c}(v)=2\}$ is connected. In this paper, we present a greedy algorithm for MinRDS with a guaranteed performance ratio at most $H(\delta _{\max }+1),$ where $H(\cdot )$ is the Harmonic number and $\delta _{\max }$ is the maximum degree of the graph. For any $\varepsilon >0,$ we show that there exists a greedy algorithm for MinCRDS with approximation ratio at most $(1+\varepsilon )\ln \delta _{\max }+O(1).$ 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.