Arbitrary Overlap Constraints in Graph Packing Problems

In earlier versions of the community discovering problem, the overlap between communities was restricted by a simple count upper-bound [17,5,11,8]. In this paper, we introduce the $\Pi$-Packing with $\alpha()$-Overlap problem to allow for more complex constraints in the overlap region than those previously studied. Let $\mathcal{V}^r$ be all possible subsets of vertices of $V(G)$ each of size at most $r$, and $\alpha: \mathcal{V}^r \times \mathcal{V}^r \to \{0,1\}$ be a function. The $\Pi$-Packing with $\alpha()$-Overlap problem seeks at least $k$ induced subgraphs in a graph $G$ subject to: (i) each subgraph has at most $r$ vertices and obeys a property $\Pi$, and (ii) for any pair $H_i,H_j$, with $i\neq j$, $\alpha(H_i, H_j) = 0$ (i.e., $H_i,H_j$ do not conflict). We also consider a variant that arises in clustering applications: each subgraph of a solution must contain a set of vertices from a given collection of sets $\mathcal{C}$, and no pair of subgraphs may share vertices from the sets of $\mathcal{C}$. In addition, we propose similar formulations for packing hypergraphs. We give an $O(r^{rk} k^{(r+1)k} n^{cr})$ algorithm for our problems where $k$ is the parameter and $c$ and $r$ are constants, provided that: i) $\Pi$ is computable in polynomial time in $n$ and ii) the function $\alpha()$ satisfies specific conditions. Specifically, $\alpha()$ is hereditary, applicable only to overlapping subgraphs, and computable in polynomial time in $n$. Motivated by practical applications we give several examples of $\alpha()$ functions which meet those conditions.