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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.02582 |
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| _version_ | 1866908934534594560 |
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| author | Fleming, Noah Hopkins, Max Yoshida, Yuichi |
| author_facet | Fleming, Noah Hopkins, Max Yoshida, Yuichi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02582 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-Signaling Locality Lower Bounds for Dominating Set Fleming, Noah Hopkins, Max Yoshida, Yuichi Data Structures and Algorithms 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. |
| title | Non-Signaling Locality Lower Bounds for Dominating Set |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.02582 |