The k-hop connected dominating set problem: approximation algorithms and hardness results
Rafael S. Coelho · 2017
Let G be a connected graph and k be a positive integer.A vertex subset D of G is a k-hop connected dominating set if the subgraph of G induced by D is connected, and for every vertex v in G, there is a vertex u in D such that the distance between v and u in G is at most k.We study the problem of finding a minimum k-hop connected dominating set of a graph (Mink-CDS).We prove that Mink-CDS is NP-hard on planar bipartite graphs of maximum degree 4. We also prove that Mink-CDS is APX-complete on bipartite graphs of maximum degree 4. We present inapproximability thresholds for Mink-CDS on bipartite and on (1, 2)-split graphs.Interestingly, one of these thresholds is a parameter of the input graph which is not a function of its number of vertices.We also discuss the complexity of computing this graph parameter.On the positive side, we show an approximation algorithm for Mink-CDS.When k = 1, we present two new approximation algorithms for the weighted version of the problem, one of them restricted to graphs with a polynomially bounded number of minimal separators.Finally, also for the weighted variant of the problem where k = 1, we discuss an integer linear programming formulation and conduct a polyhedral study of its associated polytope.