Covering with Clubs: Complexity and Approximability

Abstract

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 $O(|V{|}^{1/2-\epsilon })$, for any $\epsilon >0$, and covering $G$ with the minimum number of $3$-clubs is not approximable within factor $O(|V{|}^{1-\epsilon })$, for any $\epsilon >0$. 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.

Accepted in IWOCA 2018