Non-Signaling Locality Lower Bounds for Dominating Set

Abstract: Minimum dominating set is a basic local covering problem and a core task in distributed computing. Despite extensive study, in the classic LOCAL model there exist significant gaps between known algorithms and lower bounds. Chang and Li prove an $Ω(\log n)$-locality lower bound for a constant factor approximation, while Kuhn--Moscibroda--Wattenhofer gave an algorithm beating this bound beyond $\log Δ$-approximation, along with a weaker lower bound for this degree-dependent setting scaling roughly with $\min{\log Δ/\log\log Δ,\sqrt{\log n/\log\log n}}$. Unfortunately, this latter bound is weak for small $Δ$, and never recovers the Chang--Li bound, leaving central questions: does $O(\log Δ)$-approximation require $Ω(\log n)$ locality, and do such bounds extend beyond LOCAL? In this work, we take a major step toward answering these questions in the non-signaling model, which strictly subsumes the LOCAL, quantum-LOCAL, and bounded-dependence settings. We prove every $O(\logΔ)$-approximate non-signaling distribution for dominating set requires locality $Ω(\log n/(\logΔ\cdot \mathrm{poly}\log\logΔ))$. Further, we show for some $β\in (0,1)$, every $O(\log^βΔ)$-approximate non-signaling distribution requires locality $Ω(\log n/\logΔ)$, which combined with the KMW bound yields a degree-independent $Ω(\sqrt{\log n/\log\log n})$ quantum-LOCAL lower bound for $O(\log^βΔ)$-approximation algorithms. The proof is based on two new low-soundness sensitivity lower bounds for label cover, one via Impagliazzo--Kabanets--Wigderson-style parallel repetition with degree reduction and one from a sensitivity-preserving reworking of the Dinur--Harsha framework, together with the reductions from label cover to set cover to dominating set and the sensitivity-to-locality transfer theorem of Fleming and Yoshida.

• The paper presents a novel result showing that any O(log Δ)-approximate non-signaling algorithm for dominating set requires locality at least Ω((log n)/(log Δ · polyloglog Δ)).
• It extends lower bound results to the quantum-LOCAL model by eliminating degree-dependent factors, establishing a bound of Ω(√(log n/log log n)) for certain approximations.
• The work introduces low-soundness sensitivity-preserving reductions that blend PCP techniques with distributed computing to set new benchmarks for locality-approximation trade-offs.

Non-Signaling Locality Lower Bounds for Minimum Dominating Set

Context and Motivation

The minimum dominating set (DS) problem, a foundational local covering problem in distributed computing, seeks a minimal vertex subset such that every vertex is either in the set or adjacent to it. Its distributed complexity, especially the trade-off between locality (algorithm communication rounds) and approximation factor, remains incompletely understood, particularly in stronger distributed models. Previous work has established that, in the LOCAL model, constant-factor approximations require $\Omega(\log n)$ locality [Chang-Li, PODC'23], and that degree-dependent approximation factors can circumvent this bound with more nuanced trade-offs [Kuhn-Moscibroda-Wattenhofer, JACM'16]. However, two core open questions persist:

1. Does any $O(\log \Delta)$-approximation to DS require $\Omega(\log n)$ locality, matching what is known for constant factors?
2. Can such lower bounds be extended to the non-signaling regime, which subsumes LOCAL, quantum-LOCAL, and bounded-dependence models?

This work provides the first substantial progress by establishing new lower bounds for DS approximation in the non-signaling model, which imposes only local, causality-preserving constraints on vertex output distributions. The results both sharpen and generalize existing bounds—yielding important implications for distributed lower bounds across classical, quantum, and non-signaling settings.

Main Results

Two principal locality lower bounds for dominating set approximation in the non-signaling model are established:

• Theorem 1: Any $O(\log \Delta)$-approximate non-signaling algorithm for DS on $n$-vertex, degree-$\Delta$ graphs requires locality at least

$$\Omega\left( \frac{\log n}{\log \Delta \cdot \mathrm{polyloglog}\,\Delta} \right)$$

where the dependence on $\Delta$ only weakens the lower bound for large degrees.

• Theorem 2: For some $\beta \in (0,1)$, any $O(\log^\beta \Delta)$-approximate non-signaling DS algorithm requires locality at least

$O(\log \Delta)$0

allowing, via classic reductions, a degree-independent lower bound for quantum-LOCAL algorithms:

$O(\log \Delta)$1

for $O(\log \Delta)$2-approximation. This removes the degree-dependent term present in all classic results for the quantum-LOCAL model.

The lower bounds hold throughout the entire non-signaling regime—which strictly contains all classic and quantum distributed models of the same locality—thus applying broadly, including to models with quantum or even general bounds on correlations.

Significant technical advances include:

• Extending Chang–Li's $O(\log \Delta)$3 lower bound from constant-factor to $O(\log \Delta)$4-approximate algorithms in the non-signaling setting.
• Removing the degree-dependent term in quantum-LOCAL lower bounds for $O(\log \Delta)$5-approximation, showing the bound depends only on $O(\log \Delta)$6 for a polynomial window of approximation factors.

Implications and Technical Innovations

Non-Signaling as a Universal Lower Bound Model

These results provide unconditional lower bounds for DS approximation in any realistic distributed setting (including quantum protocols and general bounded-dependence models), as the non-signaling assumption captures locality without assuming a specific communication or physical constraint. The distinctions between the locality–approximation trade-offs for DS previously known in LOCAL and quantum-LOCAL settings are now shown to persist at the outermost boundaries of causal locality.

Low-Soundness Sensitivity Reductions

The technical centerpiece is the development of low-soundness label cover sensitivity lower bounds—a substantial generalization over earlier reductions, which were confined to the high-soundness regime. These new bounds are achieved via:

• Parallel repetition framework (Impagliazzo–Kabanets–Wigderson): Amplifies the difficulty of satisfying label cover instances to extremely low soundness regimes, bypassing bottlenecks in degree reduction.
• Sensitivity-preserving reductions: A precise transfer analysis allowing one to reduce the question of distributed locality to the perturbative sensitivity of algorithms on constraint satisfaction instances, even as soundness and degrees scale with instance size.

The paper introduces two new low-soundness sensitivity lower bounds:

Method	Approximation Factor	Locality Lower Bound	Degree/Alphabet Parameters
Parallel Repetition	$O(\log \Delta)$7	$O(\log \Delta)$8	$O(\log \Delta)$9
Dinur–Harsha PCP	$\Omega(\log n)$0	$\Omega(\log n)$1	$\Omega(\log n)$2

These results clarify previously unresolved trade-offs and obstacles in reduction chains for DS, most notably the extreme technical difficulty of constructing low-soundness, instance-local reductions suitable for downstream lifting to dominating set, covering all degrees and approximation scales.

Synthesis of PCP Reductions with Sensitivity Analysis

Through careful sensitivity-preserving PCP reductions and degree/alphabet management, the work achieves robust lower bounds throughout the entire relevant regime for distributed DS. Notably, the sensitivity lower bounds for low-soundness label cover required new forms of compositional analysis, decoding mechanisms, and the design of stability-preserving proof verifiers—a notable advance at the interface of PCP theory and distributed complexity.

Future Directions

These results leave two crucial technical frontiers:

1. Removing the $\Omega(\log n)$3 factor: The remaining gap in the $\Omega(\log n)$4 trade-off may be closed by further advances in PCP theory to reduce alphabet size without sacrificing low soundness, a central open problem (the "sliding scale" conjecture).
2. Best-of-both-worlds lower bounds: Achieving both optimal approximation ratios and tight degree-independent lower bounds may require new PCP-based innovations or combinatorial constructions to overcome current barriers in embedding/routing steps and alphabet–soundness trade-offs.

There is also the possibility to transfer the sensitivity-preserving machinery to other distributed covering and CSP-type problems, thereby obtaining robust non-signaling lower bounds in regimes not accessible via classic reductions.

Conclusion

This paper decisively advances our understanding of dominating set approximation in distributed models. By establishing strong lower bounds in the non-signaling model—thereby enveloping the classic, quantum, and causality-limited settings—it confirms that the $\Omega(\log n)$5 "barrier" for locality extends as far as current models of distributed and quantum computation permit for a rich family of approximation regimes. The innovative low-soundness sensitivity techniques developed herein should inform the design and analysis of distributed lower bounds across a range of graph covering and CSP problems.

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Reference: "Non-Signaling Locality Lower Bounds for Dominating Set" (2604.02582)