An exact algorithm for the maximum k-club problem in an undirected graph

Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

“…Table 5 summarizes our findings (see Table 3, appendix, for a full list). As first observed by Bourjolly et al [7], density 0.15 produces the hardest instances. We solve instances of these types for n = 150 typically within 2min; previous implementations needed about 6min [8], or up to an hour [17] for these instances.…”

“…Furthermore, 2-Club is also an important special case concerning the applications: For biological networks, 2-clubs and 3-clubs have been identified as the most reasonable diameter-relaxations of Clique [21], further applications of 2-Club arise in the analysis of social networks [18]. Consequently, experimental evaluations concentrate on finding 2-clubs and 3-clubs [7,8,17].…”

“…For all s ≥ 1, s-Club is NP-hard to approximate within a factor of n 1 /2− [2]; a simple approximation algorithm obtains a factor of n 1 /2 for even s ≥ 2 and a factor of n 2 /3 for odd s ≥ 3 [2]. Several heuristics [7] and integer linear programming formulations [3,7] for s-Club have been proposed and experimentally evaluated [17]. 1-Club is equivalent to Clique and thus W [1]-hard with respect to .…”

“…The above search tree strategy was introduced by Bourjolly et al [7] and has been already experimentally evaluated [8]. By simple recursion analysis the running time can be bounded by O * (α n ) where α is the golden ratio with α ≈ 1.62 [8].…”

Parameterized computational complexity of finding small-diameter subgraphs

“…s-Club is NP-complete [7], even in graphs of diameter s + 1 [3]. For s ≥ 2, every graph contains an s-club of order ∆(G)+1, where ∆(G) is the maximum degree of G, namely the graph that is induced by the closed neighborhood of a vertex with maximum degree.…”