Covering with Clubs: Complexity and Approximability

Riccardo Dondi (1) , Giancarlo Mauri (2) , Florian Sikora (3) , Italo Zoppis (2)

1 UniBg - Università degli Studi di Bergamo = University of Bergamo
2 UNIMIB - Università degli Studi di Milano-Bicocca = University of Milano-Bicocca
3 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision

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−ε), for any ε>0, and covering G with the minimum number of 3-clubs is not approximable within factor O(|V|1−ε), for any ε>0. On the positive side, we give an approximation algorithm of factor 2|V|1/2log3/2|V| for covering a graph with the minimum number of 2-clubs.

Mots clés: approximation algorithms; combinatorial optimization; graph algorithms

Domaines: Informatique [cs]

Riccardo Dondi, Giancarlo Mauri, Florian Sikora, Italo Zoppis. Covering with Clubs: Complexity and Approximability. 29th International Workshop on Combinatorial Algorithms (IWOCA 2018), Jul 2018, Singapore, Singapore. pp.153-164, ⟨10.1007/978-3-319-94667-2_13⟩. ⟨hal-02074964⟩