We study a natural generalization of the classical \textsc{Dominating Set} problem, called \textsc{Dominating Set with Quotas} (DSQ). In this problem, we are given a graph \( G \), an integer \( k \), and for each vertex \( v \in V(G) \), a lower quota \( \mathrm{lo}_v \) and an upper quota \( \mathrm{up}_v \). The goal is to determine whether there exists a set \( S \subseteq V(G) \) of size at most \( k \) such that for every vertex \( v \in V(G) \), the number of vertices in its closed neighborhood that belong to \( S \), i.e., \( |N[v] \cap S| \), lies within the range \( [\mathrm{lo}_v, \mathrm{up}_v] \). This richer model captures a variety of practical settings where both under- and over-coverage must be avoided -- such as in fault-tolerant infrastructure, load-balanced facility placement, or constrained communication networks. While DS is already known to be computationally hard, we show that the added expressiveness of per-vertex quotas in DSQ introduces additional algorithmic challenges. In particular, we prove that DSQ becomes \W[1]-hard even on structurally sparse graphs -- such as those with degeneracy 2, or excluding \( K_{3,3} \) as a subgraph -- despite these classes admitting FPT algorithms for DS. On the positive side, we show that DSQ is fixed-parameter tractable when parameterized by solution size and treewidth, and more generally, on nowhere dense graph classes. Furthermore, we design a subexponential-time algorithm for DSQ on apex-minor-free graphs using the bidimensionality framework. These results collectively offer a refined view of the algorithmic landscape of DSQ, revealing a sharp contrast with the classical DS problem and identifying the key structural properties that govern tractability.

翻译：我们研究了经典支配集（\textsc{Dominating Set}）问题的一个自然推广，称为带配额的支配集（\textsc{Dominating Set with Quotas}，简称DSQ）。在该问题中，给定一个图 \( G \)、一个整数 \( k \)，以及每个顶点 \( v \in V(G) \) 的下配额 \( \mathrm{lo}_v \) 和上配额 \( \mathrm{up}_v \)。目标是判断是否存在一个大小至多为 \( k \) 的集合 \( S \subseteq V(G) \)，使得对于每个顶点 \( v \in V(G) \)，其闭邻域中属于 \( S \) 的顶点数量，即 \( |N[v] \cap S| \)，位于区间 \( [\mathrm{lo}_v, \mathrm{up}_v] \) 内。这一更丰富的模型捕捉了多种实际场景，其中必须避免覆盖不足和覆盖过度——例如在容错基础设施、负载均衡设施选址或受限通信网络中。尽管已知支配集问题在计算上已是难解的，但我们表明，DSQ中每个顶点配额的额外表达能力引入了新的算法挑战。特别地，我们证明DSQ即使在结构稀疏的图类上（如退化度为2的图，或排除 \( K_{3,3} \) 作为子图的图）也变为\W[1]-难问题，尽管这些图类对支配集问题允许固定参数可解（FPT）算法。在正面结果方面，我们表明DSQ在以解大小和树宽为参数时是固定参数可解的，更一般地，在无无处稠密图类上也是如此。此外，我们利用双维性框架为无主小面图上的DSQ设计了一个亚指数时间算法。这些结果共同提供了对DSQ算法图景的精细视角，揭示了其与经典支配集问题的鲜明对比，并识别了决定可解性的关键结构性质。