A greedy algorithm for the minimum $$2$$ 2 -connected $$m$$ m -fold dominating set problem

To save energy and alleviate interference in a wireless sensor network, connected dominating set (CDS) has been proposed as the virtual backbone. Since nodes may fail due to accidental damage or energy depletion, it is desirable to construct a fault tolerant CDS, which can be modeled as a $$k$$ k -connected $$m$$ m -fold dominating set $$((k,m)$$ ( ( k , m ) -CDS for short): a subset of nodes $$C\subseteq V(G)$$ C ⊆ V ( G ) is a $$(k,m)$$ ( k , m ) -CDS of $$G$$ G if every node in $$V(G)\setminus C$$ V ( G ) \ C is adjacent with at least $$m$$ m nodes in $$C$$ C and the subgraph of $$G$$ G induced by $$C$$ C is $$k$$ k -connected.In this paper, we present an approximation algorithm for the minimum $$(2,m)$$ ( 2 , m ) -CDS problem with $$m\ge 2$$ m ≥ 2 . Based on a $$(1,m)$$ ( 1 , m ) -CDS, the algorithm greedily merges blocks until the connectivity is raised to two. The most difficult problem in the analysis is that the potential function used in the greedy algorithm is not submodular. By proving that an optimal solution has a specific decomposition, we managed to prove that the approximation ratio is $$\alpha +2(1+\ln \alpha )$$ α + 2 ( 1 + ln α ) , where $$\alpha $$ α is the approximation ratio for the minimum $$(1,m)$$ ( 1 , m ) -CDS problem. This improves on previous approximation ratios for the minimum $$(2,m)$$ ( 2 , m ) -CDS problem, both in general graphs and in unit disk graphs.