A dynamic programming algorithm for the maximum s-club problem on trees

Computing cliques in an undirected graph G = (V_G, E_G) is a fundamental problem in social network analysis. However, in some cases, the strict definition of a clique (a subset of vertices pairwise adjacent in G) often limits its applicability in real-world settings. To address this issue, we study the s-club: a clique relaxation that induces a subgraph of diameter at most s. Note that a clique is simply a 1-club. Computing a maximum s-club is a computationally challenging problem, as it is NP-hard for any positive integer s in arbitrary graphs. Thus, this paper presents a simple dynamic programming algorithm that efficiently computes a maximum s-club on an n-vertex tree in O(s.n) time. This algorithm outperforms existing algorithms for trees in theory and practice. This approach is a stepping stone towards computing maximum s-clubs on tree-like graphs.