Abstract: Highlights•Propose a unified concept called (ρ,ϱ)-MDS (or (ρ,ϱ)-MCDS) to describe the Minimum (Connected) Dominating Set problem and its variants.•Design and analyze a unified O(lnα)-approximation algorithm for (ρ,ϱ)-MDS and (ρ,ϱ)-MCDS, by giving a (weak) submodular potential functions.•Achieve the best approximation ratio for some specific versions of (ρ,ϱ)-MDS and (ρ,ϱ)-MCDS by the unified algorithm.