We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set. Given a red-blue bipartite graph $G$ and an auxiliary connectivity graph $G_{conn}$ on red vertices, and integers $k, t$, the task is to find a $k$-sized subset of red vertices that dominates at least $t$ blue vertices, and that induces a connected subgraph in $G_{conn}$. This formulation captures connected variants of Max Coverage, Partial Dominating Set, and Partial Vertex Cover studied in prior literature. After identifying (parameterized) inapproximability results inherited from known problems, we first show that the problem is fixed-parameter tractable by $t$. Furthermore, when the bipartite graph excludes $K_{d,d}$ as a subgraph, we design (resp. efficient) parameterized approximation schemes for approximating $t$ (resp. $k$). Notably, these FPT approximations do not impose any restrictions on $G_{conn}$. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

翻译：我们通过一个统一模型——部分连通红蓝支配集，重新审视连通性约束的覆盖问题。给定一个红蓝二分图$G$、基于红色顶点的辅助连通图$G_{conn}$以及整数$k, t$，任务是找到一个包含$k$个红色顶点的子集，使其支配至少$t$个蓝色顶点，并且在$G_{conn}$中诱导出一个连通子图。该模型涵盖了现有文献中研究的连通版最大覆盖、部分支配集和部分顶点覆盖问题。在识别出从已知问题继承的（参数化）不可近似性结果后，我们首先证明该问题对于参数$t$是固定参数可解的。进一步地，当二分图不包含$K_{d,d}$作为子图时，我们设计了近似$t$（相应地，近似$k$）的（高效）参数化近似方案。值得注意的是，这些FPT近似算法不对$G_{conn}$施加任何限制。这些结果共同划定了连通性约束覆盖问题的困难性与FPT可近似性之间的边界。