Kernelization Algorithms for Packing Problems Allowing Overlaps

Abstract

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 $\mathcal {S}' \subseteq \mathcal {S}$ 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 $r$-Set Packing with $t$-Overlap each pair of sets in $\mathcal {S'}$ overlaps in at most $t$ elements. For both problems, 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 $H \in \mathcal {H}$ where each member of $\mathcal {H}$ has at most $r$ vertices. In $\mathcal {H}$-Packing with $t$-Membership, each vertex of $G$ belongs to at most $t$ subgraphs of $\mathcal {K}$ while in $\mathcal {H}$-Packing with $t$-Overlap each pair of subgraphs in $\mathcal {K}$ overlaps in at most $t$ vertices.

Here, we show NP-Completeness results for all of our packing problems. Furthermore, we give a dichotomy result for $\mathcal {H}$-Packing with $t$- Membership analogous to the Kirkpatrick and Hell [12]. Given this intractability, we reduce $r$-Set Packing with $t$-Membership and $t$-Overlap to problem kernels with $O((r+1)^r k^{r})$ and $O(r^r k^{r-t-1})$ elements, respectively. Similarly, we reduce $\mathcal {H}$-Packing with $t$-Membership and $t$-Overlap to instances with $O((r+1)^r k^{r})$ and $O(r^r k^{r-t-1})$ vertices, respectively. In all cases, $k$ is the input parameter while $t$ and $r$ are constants.