Non-Signaling Locality Lower Bounds for Dominating Set

TL;DR

This paper establishes new non-signaling locality lower bounds for approximating the dominating set problem, extending understanding beyond the LOCAL model into quantum and non-signaling frameworks.

Contribution

It proves degree-independent lower bounds for non-signaling distributions approximating dominating sets, advancing the theoretical understanding of distributed computing limits.

Findings

Every $O( ext{log} \Delta)$-approximate non-signaling distribution requires $ ext{Omega}( ext{log} n/( ext{log}\Delta ext{· polyloglog} ext ext{log} ))$ locality.

For some $eta ext ext{in}(0,1)$, $O( ext{log}^eta ext ext{Delta})$-approximate distributions require $ ext{Omega}( ext{log} n/ ext{log} ext ext{Delta})$ locality.

Derived a degree-independent quantum-LOCAL lower bound of $ ext{Omega}( ext ext{sqrt}( ext ext{log} n/ ext ext{log} ext ext{log} ext ext{log} ext ext{n}))$.

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 $\Omega(\log n)$-locality lower bound for a constant factor approximation, while Kuhn--Moscibroda--Wattenhofer gave an algorithm beating this bound beyond $\log \Delta$-approximation, along with a weaker lower bound for this degree-dependent setting scaling roughly with $\min\{\log \Delta/\log\log \Delta,\sqrt{\log n/\log\log n}\}$. Unfortunately, this latter bound is weak for small $\Delta$, and never recovers the Chang--Li bound, leaving central questions: does $O(\log \Delta)$-approximation require $\Omega(\log n)$ 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 $O(\log\Delta)$-approximate non-signaling distribution for dominating set requires locality $\Omega(\log n/(\log\Delta \cdot \mathrm{poly}\log\log\Delta))$. Further, we show for some $\beta \in (0,1)$, every $O(\log^\beta \Delta)$-approximate non-signaling distribution requires locality $\Omega(\log n/\log\Delta)$, which combined with the KMW bound yields a degree-independent $\Omega(\sqrt{\log n/\log\log n})$ quantum-LOCAL lower bound for $O(\log^\beta\Delta)$-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.