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 $t=1$.

翻译：网络稀疏化是指减少给定图的边数同时保留某些关键图性质的任务。在社区感知网络稀疏化中，保留的性质涉及由图的社区（作为顶点子集给出）诱导的子图。该问题形式化为$\Pi$-网络稀疏化问题：给定一个边加权图$G$，一个由$c$个$V(G)$子集（社区）构成的集合$Z$，以及两个数$\ell$和$b$，问题在于是否存在$G$的一个生成子图$G'$，其边数不超过$\ell$且总权重不超过$b$，使得对于每个社区$C$，$G'[C]$满足$\Pi$。这里我们考虑两种图性质$\Pi$：连通性性质（连通性NWS）和具有生成星的性质（星形NWS）。由于这两个问题均为NP难问题，我们研究它们的参数化与精细复杂度。基于ETH，我们为两个问题提供了$2^{\Omega(n^2+c)} poly(n+|Z|)$时间运行时间的严格下界，其中$n$是$G$的顶点数。即使当所有社区大小至多为4、$G$为完全图且每条边权重均为单位权重时，该下界仍然成立。对于连通性性质，$G$为完全图的单位权重情形即为众所周知的用最少边数计算超图支撑问题。随后，我们研究了两个问题在解图$G'$的反馈边数$t$参数化下的复杂度。对于星形NWS，我们提出了一个关于$t$的XP算法。这回答了Korach和Stern [Disc. Appl. Math. '08] 提出的关于当$t=0$时是否存在多项式时间算法的公开问题。相比之下，我们证明连通性NWS在$t=1$时是NP难的，表明已知的当$t=0$时的多项式时间算法 [Korach and Stern, Math. Program. '03; Klemz et al., SWAT '14] 无法被推广。