  Given a graph of interactions, a module (also called a community or cluster)
is a subset of nodes whose fitness is a function of the statistical
significance of the pairwise interactions of nodes in the module. The topic of
this paper is a model-based community finding approach, commonly referred to as
modularity clustering, that was originally proposed by Newman and has
subsequently been extremely popular in practice. Various heuristic methods are
currently employed for finding the optimal solution. However, the exact
computational complexity of this approach is still largely unknown.
  To this end, we initiate a systematic study of the computational complexity
of modularity clustering. Due to the specific quadratic nature of the
modularity function, it is necessary to study its value on sparse graphs and
dense graphs separately. Our main results include a (1+\eps)-inapproximability
for dense graphs and a logarithmic approximation for sparse graphs. We make use
of several combinatorial properties of modularity to get these results. These
are the first non-trivial approximability results beyond the previously known
NP-hardness results.
