Kernelization Algorithms for Packing Problems Allowing Overlaps (Extended Version)

Abstract: We consider the problem of discovering overlapping communities in networks which we model as generalizations of Graph Packing problems with overlap. We seek a collection $\mathcal{S}&#39; \subseteq \mathcal{S}$ consisting of at least $k$ sets subject to certain disjointness restrictions. In the $r$-Set Packing with $t$-Membership, each element of $\mathcal{U}$ belongs to at most $t$ sets of $\mathcal{S&#39;}$ while in $t$-Overlap each pair of sets in $\mathcal{S&#39;}$ overlaps in at most $t$ elements. Each set of $\mathcal{S}$ has at most $r$ elements. Similarly, both of our graph packing problems seek a collection $\mathcal{K}$ of at least $k$ subgraphs in a graph $G$ each isomorphic to a graph $H \in \mathcal{H}$. In $\mathcal{H}$-Packing with $t$-Membership, each vertex of $G$ belongs to at most $t$ subgraphs of $\mathcal{K}$ while in $t$-Overlap each pair of subgraphs in $\mathcal{K}$ overlaps in at most $t$ vertices. Each member of $\mathcal{H}$ has at most $r$ vertices and $m$ edges. We show NP-Completeness results for all of our packing problems and we give a dichotomy result for the $\mathcal{H}$-Packing with $t$-Membership problem analogous to the Kirkpatrick and Hell \cite{Kirk78}. We reduce the $r$-Set Packing with $t$-Membership to a problem kernel with $O((r+1)^r k<sup>{r})$ elements while we achieve a kernel with $O(r^r k<sup>{r-t-1})$ elements for the $r$-Set Packing with $t$-Overlap. In addition, we reduce the $\mathcal{H}$-Packing with $t$-Membership and its edge version to problem kernels with $O((r+1)^r k<sup>{r})$ and $O((m+1)^{m} k<sup>{{m}})$ vertices, respectively. On the other hand, we achieve kernels with $O(r^r k<sup>{r-t-1})$ and $O(m^{m} k<sup>{m-t-1})$ vertices for the $\mathcal{H}$-Packing with $t$-Overlap and its edge version, respectively. In all cases, $k$ is the input parameter while $t$, $r$, and $m$ are constants.