Dominating Set with Quotas: Balancing Coverage and Constraints

2026-04-06

AI Summary

This study addresses the Dominating Set with Quotas (DSQ) problem, which seeks a vertex subset of size at most \(k\) such that each vertex is dominated by a number of selected vertices within prescribed lower and upper bounds in its closed neighborhood, thereby balancing coverage quality and load distribution. By leveraging parameterized algorithms, treewidth decomposition, and nowhere-dense graph theory, the work demonstrates that DSQ exhibits significantly higher computational complexity than the classical Dominating Set problem on sparse graphs. Specifically, it proves that DSQ is W[1]-hard even on graphs of degeneracy two or those excluding \(K_{3,3}\) as a subgraph. Moreover, the paper establishes the first fixed-parameter tractability results for DSQ with respect to both treewidth and solution size, and presents a subexponential-time algorithm on apex-minor-free graphs.

Abstract

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.

Problem

• Dominating Set
• Quotas
• Graph Algorithms
• Parameterized Complexity
• Coverage Constraints

Innovation

• Dominating Set with Quotas
• fixed-parameter tractability
• nowhere dense graphs
• bidimensionality
• W[1]-hardness