IWOCA 2018: Combinatorial Algorithms pp 153-164

# Covering with Clubs: Complexity and Approximability

• Riccardo Dondi
• Giancarlo Mauri
• Florian Sikora
• Italo Zoppis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)

## 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.

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## Authors and Affiliations

• Riccardo Dondi
• 1
• Giancarlo Mauri
• 2
• Florian Sikora
• 3
Email author
• Italo Zoppis
• 2
1. 1.Università degli Studi di BergamoBergamoItaly
2. 2.Università degli Studi di Milano-BicoccaMilanItaly
3. 3.Université Paris-Dauphine, PSL Research University, CNRS UMR 7243, LAMSADEParisFrance