Covering with Clubs: Complexity and Approximability

Finding cohesive subgraphs in a network is a well-known problem in graph theory. Several alternative formulations of cohesive subgraph have been proposed, a notable example being s-club, which is a subgraph where each vertex is at distance at most s to the others. Here we consider the problem of covering a given graph with the minimum number of s-clubs. We study the computational and approximation complexity of this problem, when s is equal to 2 or 3. First, we show that deciding if there exists a cover of a graph with three 2-clubs is NP-complete, and that deciding if there exists a cover of a graph with two 3-clubs is NP-complete. Then, we consider the approximation complexity of covering a graph with the minimum number of 2-clubs and 3-clubs. We show that, given a graph \(G=(V,E)\) to be covered, covering G with the minimum number of 2-clubs is not approximable within factor \(O(|V|^{1/2 -\varepsilon })\), for any \(\varepsilon >0\), and covering G with the minimum number of 3-clubs is not approximable within factor \(O(|V|^{1 -\varepsilon })\), for any \(\varepsilon >0\). On the positive side, we give an approximation algorithm of factor \(2|V|^{1/2}\log ^{3/2} |V|\) for covering a graph with the minimum number of 2-clubs.