Covering with Clubs: Complexity and Approximability

TL;DR

This paper investigates the computational complexity and approximability of covering graphs with a minimal number of s-clubs, revealing NP-completeness results and providing approximation algorithms for s=2 and s=3.

Contribution

It establishes NP-completeness for covering with two 3-clubs and three 2-clubs, and offers new approximation algorithms with specific bounds.

Findings

Covering with three 2-clubs is NP-complete.

Covering with two 3-clubs is NP-complete.

Provided an approximation algorithm with factor 2|V|^{1/2} log^{3/2} |V| for 2-club covering.

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 $O(|V|^{1/2 -\varepsilon})$, 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.