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. In this paper, we introduce the II-Packing with alpha()-Overlap problem to allow for more complex constraints in the overlap region than those previously studied. Let V-r be all possible subsets of vertices of V(G) each of size at most r, and alpha : V-r x V-r -> {0, 1} be a function. The II-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 not equal j, alpha(H-i, H-j) = 0 (i.e., the pair H-i, H-j does 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 C, and no pair of subgraphs may share vertices from the sets of C. In addition, we propose similar formulations for packing hypergraphs. We give an O(r(rk)k((r+1)k)n(r)) 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 and r. Motivated by practical applications we give several examples of a() functions which meet those conditions.