Arbitrary Overlap Constraints in Graph Packing Problems
Abstract
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
Π
-Packing with
α()
-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
α:
V
r
×
V
r
→{0,1}
be a function. The
Π
-Packing with
α()
-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
Π
, and (ii) for any pair
H
i
,
H
j
, with
i≠j
,
α(
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
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
cr
)
algorithm for our problems where
k
is the parameter and
c
and
r
are constants, provided that: i)
Π
is computable in polynomial time in
n
and ii) the function
α()
satisfies specific conditions. Specifically,
α()
is hereditary, applicable only to overlapping subgraphs, and computable in polynomial time in
n
. Motivated by practical applications we give several examples of
α()
functions which meet those conditions.