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Non-Signaling Locality Lower Bounds for Dominating Set

Noah Fleming, Max Hopkins, Yuichi Yoshida

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.

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 -locality lower bound for a constant factor approximation, while Kuhn--Moscibroda--Wattenhofer gave an algorithm beating this bound beyond -approximation, along with a weaker lower bound for this degree-dependent setting scaling roughly with . Unfortunately, this latter bound is weak for small , and never recovers the Chang--Li bound, leaving central questions: does -approximation require 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 -approximate non-signaling distribution for dominating set requires locality . Further, we show for some , every -approximate non-signaling distribution requires locality , which combined with the KMW bound yields a degree-independent quantum-LOCAL lower bound for -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.

Paper Structure

This paper contains 92 sections, 53 theorems, 539 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proof Overview
  3. Transfer principle.
  4. Reduction chain.
  5. Key Ingredients.
  6. The Challenge of Maintaining Sensitivity.
  7. Proof Overview of \ref{['thm:intro-label-cover-sensitivity-IKW']}
  8. Combinatorial Parallel Repetition.
  9. Imbalanced Graphs and Sensitivity.
  10. The Recovery Procedure.
  11. Degree Reduction
  12. Proof Overview of \ref{['thm:intro-label-cover']}
  13. Dinur--Harsha Composition.
  14. Stage-$0$: Keeping Clause-Swaps Local.
  15. Stable Composition: Padding the Graph.
  16. ...and 77 more sections

Key Result

Theorem 1.1

There exists an integer $\Delta_0\ge 2$ and an absolute constant $C>0$ such that for all sufficiently large $n$ and every integer $\Delta \geq \Delta_0$, any $C \log \Delta$-approximate non-signaling distribution for the dominating set problem on $n$-vertex graphs with maximum degree at most $\Delta $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Before and after applying degree-reduction on a right vertex $v$ with graph $H_v$.
  • Figure 2: The stage-$0$ left-predicate label cover instance.
  • Figure 3: The transformation of the alphabet reduction procedure ${\cal T}_{AR}$ on one edge.
  • Figure 4: The conversion of a vertex $v$ and its neighborhood under composition.
  • Figure :

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4: Label Cover
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1: Left-predicate label cover instance
  • Definition 2.2: Earth mover's distance
  • Lemma 2.3: Neighboring witness extraction
  • proof
  • ...and 109 more