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It does not matter how you define locally checkable labelings

Antonio Cruciani, Avinandan Das, Alesya Raevskaya, Jukka Suomela

TL;DR

This work presents a very restricted family of locally checkable problems (essentially, the"node-edge checkable"formalism familiar from round elimination, restricted to regular unlabeled graphs); most importantly, such problems cannot directly refer to e.g. the existence of short cycles.

Abstract

Locally checkable labeling problems (LCLs) form the foundation of the modern theory of distributed graph algorithms. First introduced in the seminal paper by Naor and Stockmeyer [STOC 1993], these are graph problems that can be described by listing a finite set of valid local neighborhoods. This seemingly simple definition strikes a careful balance between two objectives: they are a family of problems that is broad enough so that it captures numerous problems that are of interest to researchers working in this field, yet restrictive enough so that it is possible to prove strong theorems that hold for all LCL problems. In particular, the distributed complexity landscape of LCL problems is now very well understood. In this work we show that the family of LCL problems is extremely robust to variations. We present a very restricted family of locally checkable problems (essentially, the "node-edge checkable" formalism familiar from round elimination, restricted to regular unlabeled graphs); most importantly, such problems cannot directly refer to e.g. the existence of short cycles. We show that one can translate between the two formalisms (there are local reductions in both directions that only need access to a symmetry-breaking oracle, and hence the overhead is at most an additive $O(\log^* n)$ rounds in the LOCAL model).

It does not matter how you define locally checkable labelings

TL;DR

This work presents a very restricted family of locally checkable problems (essentially, the"node-edge checkable"formalism familiar from round elimination, restricted to regular unlabeled graphs); most importantly, such problems cannot directly refer to e.g. the existence of short cycles.

Abstract

Locally checkable labeling problems (LCLs) form the foundation of the modern theory of distributed graph algorithms. First introduced in the seminal paper by Naor and Stockmeyer [STOC 1993], these are graph problems that can be described by listing a finite set of valid local neighborhoods. This seemingly simple definition strikes a careful balance between two objectives: they are a family of problems that is broad enough so that it captures numerous problems that are of interest to researchers working in this field, yet restrictive enough so that it is possible to prove strong theorems that hold for all LCL problems. In particular, the distributed complexity landscape of LCL problems is now very well understood. In this work we show that the family of LCL problems is extremely robust to variations. We present a very restricted family of locally checkable problems (essentially, the "node-edge checkable" formalism familiar from round elimination, restricted to regular unlabeled graphs); most importantly, such problems cannot directly refer to e.g. the existence of short cycles. We show that one can translate between the two formalisms (there are local reductions in both directions that only need access to a symmetry-breaking oracle, and hence the overhead is at most an additive rounds in the LOCAL model).
Paper Structure (97 sections, 34 theorems, 45 equations, 2 figures)

This paper contains 97 sections, 34 theorems, 45 equations, 2 figures.

Table of Contents

  1. Abstract.
  2. Introduction
  3. What are LCL problems?
  4. Why do we study LCL problems?
  5. Success stories.
  6. Recent counterintuitive findings.
  7. Key features that artificial problems exploit.
  8. RE formalism---a concrete candidate definition of restricted LCLs.
  9. Our contribution: RE formalism $\approx$ general LCLs.
  10. Corollaries of our work.
  11. Discussion.
  12. Future work and open questions.
  13. Proof overview
  14. $A \to B$: making it regular and input-free.
  15. $B \to C$: making it PN-checkable.
  16. ...and 82 more sections

Key Result

Lemma 3.1

Fix an arbitrary node $v\in V(G')$ such that it is not a part of any edge gadget. Let $(G", \sigma^{\operatorname{}}_{V(G")}, \sigma^{\operatorname{}}_{E(G")})$ be the decoding of $B^{\Lambda}_{G'}(v)$. Let $u\in V(G)$ and $u'\in V(G")$ be the decoded nodes such that $v$ belongs to both $\mathsf{dec

Figures (2)

  • Figure 1: Illustrations for gadget expansion
  • Figure 2: Illustrations for \ref{['lem:outer_disconnected']}. Issues highlighted in pink, potential placements of 1 are in blue and implied placements of 3 and 4 are in orange.

Theorems & Definitions (66)

  • Remark
  • Remark
  • Remark
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3: LOCAL Decoding
  • proof
  • Lemma 3.4: Locality of encoding
  • ...and 56 more