  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 $\mathcal{S}' \subseteq \mathcal{S}$ 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 $\mathcal{S'}$ while in $t$-Overlap each pair of sets in
$\mathcal{S'}$ 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 $O((r+1)^r k^{r})$
elements while we achieve a kernel with $O(r^r k^{r-t-1})$ 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 $O((r+1)^r k^{r})$ and $O((m+1)^{m} k^{{m}})$ vertices,
respectively. On the other hand, we achieve kernels with $O(r^r k^{r-t-1})$ and
$O(m^{m} k^{m-t-1})$ 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.
