The G-Packing with t-Overlap Problem

We introduce the k-G-Packing with t-Overlap problem to formalize the problem of finding communities in a network. In the k-G-Packing with t-Overlap problem, we search for at least k communities with possible overlap. In contrast with previous work where communities are disjoint, we regulate the overlap through a parameter t. Our focus is the parameterized complexity of the k-G-Packing with t-Overlap problem. Here, we provide a new technique for this problem generalizing the crown decomposition technique [ 2]. Using our global rule, we achieve a kernel with size bounded by 2(rk-r) for the k-G-Packing with t-Overlap problem when t = r - 2 and G is a clique of size r.