  Network sparsification is the task of reducing the number of edges of a given
graph while preserving some crucial graph property. In community-aware network
sparsification, the preserved property concerns the subgraphs that are induced
by the communities of the graph which are given as vertex subsets. This is
formalized in the $\Pi$-Network Sparsification problem: given an edge-weighted
graph $G$, a collection $Z$ of $c$ subsets of $V(G)$ (communities), and two
numbers $\ell, b$, the question is whether there exists a spanning subgraph
$G'$ of $G$ with at most $\ell$ edges of total weight at most $b$ such that
$G'[C]$ fulfills $\Pi$ for each community $C$. Here, we consider two graph
properties $\Pi$: the connectivity property (Connectivity NWS) and the property
of having a spanning star (Stars NWS). Since both problems are NP-hard, we
study their parameterized and fine-grained complexity.
  We provide a tight $2^{\Omega(n^2+c)} poly(n+|Z|)$-time running time lower
bound based on the ETH for both problems, where $n$ is the number of vertices
in $G$. The lower bound holds even in the restricted case when all communities
have size at most 4, $G$ is a clique, and every edge has unit weight. For the
connectivity property, the unit weight case with $G$ being a clique is the
well-studied problem of computing a hypergraph support with a minimum number of
edges. We then study the complexity of both problems parameterized by the
feedback edge number $t$ of the solution graph $G'$. For Stars NWS, we present
an XP-algorithm for $t$. This answers an open question by Korach and Stern
[Disc. Appl. Math. '08] who asked for the existence of polynomial-time
algorithms for $t=0$. In contrast, we show for Connectivity NWS that known
polynomial-time algorithms for $t=0$ [Korach and Stern, Math. Program. '03;
Klemz et al., SWAT '14] cannot be extended by showing that Connectivity NWS is
NP-hard for $t=1$.
