Non-Signaling Locality Lower Bounds for Dominating Set

2026-04-02

AI Summary

This study addresses the locality lower bounds for approximating minimum dominating sets in distributed computing, focusing on whether an $O(\log \Delta)$-approximation inherently requires $\Omega(\log n)$ communication rounds. By establishing a novel connection between dominating sets and label cover problems in a generalized non-signaling model and introducing a new sensitivity-based lower bound technique, the work proves that any $O(\log \Delta)$-approximation algorithm necessitates $\Omega\left(\frac{\log n}{\log \Delta \cdot \mathrm{poly}\log\log \Delta}\right)$ locality, while an $O(\log^\beta \Delta)$-approximation incurs an $\Omega\left(\frac{\log n}{\log \Delta}\right)$ lower bound. Furthermore, it derives a degree-independent $\sqrt{\frac{\log n}{\log \log n}}$ lower bound in the quantum LOCAL model. These results bridge a long-standing gap between upper and lower bounds in the classical LOCAL model and extend to quantum and bounded-independence settings.

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.

Problem

• dominating set
• locality lower bounds
• non-signaling model
• distributed approximation
• LOCAL model

Innovation

• non-signaling locality
• dominating set
• label cover
• distributed lower bounds
• quantum-LOCAL