Kernelization Algorithms for Packing Problems Allowing Overlaps (Extended Version)

We consider the problem of discovering overlapping communities in networks which we model as generalizations of the Set and Graph Packing problems with overlap. As usual for Set Packing problems we seek a collection S consisting of at least k sets subject to certain disjointness restrictions. In an r-Set Packing with t-Membership, each element of U belongs to at most t sets while in an r-Set Packing with t-Overlap each pair of sets overlaps in at most t elements. For both problems, each set of S has at most r elements, and t and r are constants. Similarly, both of our graph packing problems seek a collection of at least k subgraphs in a graph G each isomorphic to a graph H ∈ H. In an H-Packing with t-Membership, each vertex of G belongs to at most t subgraphs while in an H-Packing with t-Overlap each pair of subgraphs overlaps in at most t vertices. For both problems, each member of H has at most r vertices and m edges, where t, r, and m are constants. Here, we show NP-Completeness results for all of our packing problems. Furthermore, we give a dichotomy result for the H-Packing with t-Membership problem analogous to the Kirkpatrick and Hell [16]. Given this intractability, we reduce the r-Set Packing with t-Overlap to an O(rk) problem kernel while we achieve an O(rk) problem kernel for the r-Set Packing with t-Membership. In addition, we reduce the H-Packing with t-Membership and its edge version to O(rk) and O(mk) kernels, respectively. On the other hand, we achieve O(rk) andO(mk) kernels for theH-Packing with t-Overlap and its edge version, respectively. Our kernel results match the best kernel for Set and Graph Packing.