Breaking the O(ln n) Barrier: An Enhanced Approximation Algorithm for Fault-Tolerant Minimum Weight Connected Dominating Set

Finding a connected dominating set (CDS) in a given graph is a fundamental problem and has been studied intensively for a long time because of its application in computer science and operations research, e.g., connected facility location and wireless networks. In some cases, fault-tolerance is desirable. Taking wireless networks as an example, since wireless nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone has some fault-tolerance. Such a problem can be modeled as finding a minimum k-connected m-fold dominating set ((k, m)-CDS) of a graph G = (V, E), which is a node set D such that every node outside of D has at least m neighbors in D and the subgraph of G induced by D is k-connected. In this paper, we study the minimum weight (1, m)-CDS problem ((1, m)-MWCDS), and present an (H(δ + m) + 2H(δ − 1))-approximation algorithm, where δ is the maximum degree of the graph and H(·) is the Harmonic number. Notice that the state-of-the-art algorithm achieves O(l...