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arxiv: 2604.04912 · v1 · submitted 2026-04-06 · 💻 cs.DS

Dominating Set with Quotas: Balancing Coverage and Constraints

Sobyasachi Chatterjee , Sushmita Gupta , Saket Saurabh , Sanjay Seetharaman , Anannya Upasana This is my paper

Pith reviewed 2026-05-10 19:20 UTC · model grok-4.3

classification 💻 cs.DS
keywords dominating setquotasW[1]-hardnessfixed-parameter tractabilitynowhere dense graphsapex-minor-free graphsbidimensionalitydegeneracy
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The pith

Adding quotas to dominating set makes it W[1]-hard on degeneracy-2 graphs, where the classical version is FPT.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies Dominating Set with Quotas, a variant requiring a set S of size at most k such that for every vertex the number of selected vertices in its closed neighborhood falls between given lower and upper bounds. It shows this version is W[1]-hard even on graphs of degeneracy 2 and on graphs excluding K_{3,3} as a subgraph, classes that admit FPT algorithms for ordinary dominating set. On the tractable side, DSQ is fixed-parameter tractable when parameterized by solution size together with treewidth, and remains FPT on all nowhere dense graph classes. It also admits a subexponential-time algorithm on apex-minor-free graphs derived via the bidimensionality framework.

Core claim

DSQ is W[1]-hard even on graphs with degeneracy 2 or excluding K_{3,3} as a subgraph, yet DSQ is FPT when parameterized by solution size and treewidth, FPT on nowhere dense classes, and has a subexponential-time algorithm on apex-minor-free graphs.

What carries the argument

The per-vertex lower and upper quota bounds on the coverage count |N[v] ∩ S| for each vertex v, enforced simultaneously with the global size bound |S| ≤ k.

Load-bearing premise

The hardness reductions can introduce the quota numbers into the input without increasing the degeneracy or creating forbidden minors in the target graph classes.

What would settle it

A fixed-parameter algorithm running in f(k) n^c time on all degeneracy-2 graphs would falsify the W[1]-hardness result for that class.

Figures

Figures reproduced from arXiv: 2604.04912 by Anannya Upasana, Saket Saurabh, Sanjay Seetharaman, Sobyasachi Chatterjee, Sushmita Gupta.

Figure 1
Figure 1. Figure 1: The instance J 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. • For each vertex of the form y i uv, at most one of x i u and x i v is in S ′ since xM ∈ S ′ . • For each i ∈ [r], we have ℓ ≤ |S ′ ∩ {x i v : v ∈ V (H)}| ≤ h. Thus, we obtain col: V (H) → [r], by mapping to each vertex v ∈ V (H) the unique color i… view at source ↗
Figure 2
Figure 2. Figure 2: The instance J constructed in the reduction from IS Lemma 3.1. I is a yes-instance of IS if and only if J is a yes-instance of DSQ. Proof. Suppose that S = {vi}i∈[r] is an independent set in I. We claim that S ′ = {xvi }i∈[r] is a solution in J . We prove it by showing that each vertex in V (G) is properly dominated by S ′ . • The pendant p α (with quota ⟨0, 0⟩) is not dominated. • The vertex α (with quota… view at source ↗
Figure 3
Figure 3. Figure 3: The gadget for a vertex v with quota ⟨1, 3⟩ in an instance where f(0)= 4 writing an FO formula based on that to solve DSQ1 . But this results in different vocabularies for different values of k. We present an algorithm using only the edge relation in the vocabulary that decides whether G has a DSQ of size at most k. For each vertex v, we attach a quota gadget to it: (v) − (Kf(0)) − (path on flq(v) + 1 vert… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces Dominating Set with Quotas (DSQ), a generalization of Dominating Set in which each vertex v is equipped with lower and upper quotas lo_v and up_v; the task is to decide whether there exists a set S of size at most k such that for every v the value |N[v] ∩ S| lies in [lo_v, up_v]. The authors prove that DSQ is W[1]-hard even on degeneracy-2 graphs and on K_{3,3}-free graphs (classes on which ordinary Dominating Set is FPT), while showing that DSQ is FPT when parameterized by solution size plus treewidth, FPT on nowhere-dense classes, and solvable in subexponential time on apex-minor-free graphs via bidimensionality.

Significance. If the stated hardness and algorithmic results hold, the work supplies a precise delineation of how the addition of per-vertex coverage quotas alters the parameterized complexity of domination problems on sparse graphs. The contrast with classical DS on degeneracy-2 and K_{3,3}-free graphs is the most striking contribution, and the positive results correctly reuse standard tools (treewidth DP, nowhere-dense algorithms, bidimensionality) without introducing new ad-hoc parameters.

minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from an explicit statement of the range of the quota values (e.g., whether they are bounded by a function of k or can be arbitrary integers up to n).
  2. [Introduction] A short comparison table contrasting the complexity of DS versus DSQ on the same graph classes would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of our work and for highlighting the significance of the contrast between DSQ and classical Dominating Set on degeneracy-2 and K_{3,3}-free graphs. We are pleased that the positive results are viewed as correctly reusing standard tools. No specific major comments were provided in the report, so we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity; results via new reductions and standard techniques

full rationale

The paper establishes W[1]-hardness for DSQ via explicit reductions on degeneracy-2 and K_{3,3}-free graphs, then shows FPT membership by k+tw, on nowhere-dense classes, and subexponential time on apex-minor-free graphs via bidimensionality. These steps rely on new constructions and applications of existing parameterized machinery rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The derivation chain is self-contained against external benchmarks in parameterized complexity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters, invented entities, or ad-hoc axioms; all results rest on the standard definitions and machinery of parameterized complexity theory and structural graph theory.

axioms (1)
  • standard math Standard definitions of W[1]-hardness, FPT, treewidth, degeneracy, nowhere dense classes, and apex-minor-free graphs.
    These are the background notions used to state both the hardness and the algorithmic results.

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Reference graph

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