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Dominating Set with Quotas: Balancing Coverage and Constraints

Sobyasachi Chatterjee, Sushmita Gupta, Saket Saurabh, Sanjay Seetharaman, Anannya Upasana

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.

Dominating Set with Quotas: Balancing Coverage and Constraints

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 , an integer , and for each vertex \( v \in V(G) \), a lower quota and an upper quota . The goal is to determine whether there exists a set \( S \subseteq V(G) \) of size at most such that for every vertex \( v \in V(G) \), the number of vertices in its closed neighborhood that belong to , i.e., , lies within the range . 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 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.

Paper Structure

This paper contains 18 sections, 12 theorems, 12 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Preliminaries
  5. Parameterized Complexity.
  6. Minors.
  7. Treewidth.
  8. DSQ on graphs of bounded treewidth
  9. Hardness of DSQ parameterized by treewidth
  10. Algorithms
  11. On computing the entries at join nodes
  12. DSQ on graphs of bounded degeneracy
  13. DSQ on nowhere dense graphs
  14. FO model-checking.
  15. DSQ on apex-minor-free graphs
  16. ...and 3 more sections

Key Result

proposition 1

Given an equitable coloring in a graph with $n$ vertices using $r$ colors, each color class is of size either $\lfloor n/r \rfloor$ or $\lceil n/r \rceil$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: The instance $\mathcal{J}\xspace$ constructed in the reduction from E-EC. One can view the construction as $r$ verticals: one for each color and $n$ horizontals: one for each vertex.
  • Figure 2: The instance $\mathcal{J}\xspace$ constructed in the reduction from IS
  • Figure 3: The gadget for a vertex $v$ with quota $\langle 1,3 \rangle$ in an instance where $f(0)\!=\!4$

Theorems & Definitions (24)

  • proposition 1
  • proof
  • proof
  • proof
  • theorem 2.1
  • theorem 2.2
  • proof
  • lemma 1
  • proof
  • theorem 3.1
  • ...and 14 more