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New Hardness Results for the LOCAL Model via a Simple Self-Reduction

Alkida Balliu, Filippo Casagrande, Francesco d'Amore, Dennis Olivetti

TL;DR

This paper advances the understanding of LOCAL model hardness by presenting a simplified self-reduction variant of round elimination and applying it to fundamental problems. It proves lower bounds for maximal $b$-matching and for $( abla + k)$-edge coloring, with explicit rounds depending on Δ and n, and recovers the maximal-matching bound when $b=1$ as a corollary. The approach centers on a node-centric, self-contained framework that avoids heavier constructs, and extends to trees via lifting arguments. The results enhance the toolkit for distributed lower bounds and illuminate the role of self-reduction in deriving tight, generalizable bounds.

Abstract

Very recently, Khoury and Schild [FOCS 2025] showed that any randomized LOCAL algorithm that solves maximal matching requires $Ω(\min\{\log Δ, \log_Δn\})$ rounds, where $n$ is the number of nodes in the graph and $Δ$ is the maximum degree. This result is shown through a new technique, called round elimination via self-reduction. The lower bound proof is beautiful and presents very nice ideas. However, it spans more than 25 pages of technical details, and hence it is hard to digest and generalize to other problems. Historically, the simplification of proofs and techniques has marked an important turning point in our understanding of the complexity of graph problems. Our paper makes a step forward towards this direction, and provides the following contributions. 1. We present a short and simplified version of the round elimination via self-reduction technique. The simplification of this technique enables us to obtain the following two hardness results. 2. We show that any randomized LOCAL algorithm that solves the maximal $b$-matching problem requires $Ω(\min\{\log_{1+b}Δ, \log_Δn\})$ and $Ω(\sqrt{\log_{1+b} n})$ rounds. We recall that the $b$-matching problem is a generalization of the matching problem where each vertex can have up to $b$ incident edges in the matching. As a corollary, for $b=1$, we obtain a short proof for the maximal matching lower bound shown by Khoury and Schild. 3. Finally, we show that any randomized LOCAL algorithm that properly colors the edges of a graph with $Δ+ k$ colors requires $Ω(\min\{\log Δ, \log_Δn\})$ and $Ω(\sqrt{\log n})$ rounds, for any $k\le Δ^{1-\varepsilon}$ and any constant $\varepsilon > 0$.

New Hardness Results for the LOCAL Model via a Simple Self-Reduction

TL;DR

This paper advances the understanding of LOCAL model hardness by presenting a simplified self-reduction variant of round elimination and applying it to fundamental problems. It proves lower bounds for maximal -matching and for -edge coloring, with explicit rounds depending on Δ and n, and recovers the maximal-matching bound when as a corollary. The approach centers on a node-centric, self-contained framework that avoids heavier constructs, and extends to trees via lifting arguments. The results enhance the toolkit for distributed lower bounds and illuminate the role of self-reduction in deriving tight, generalizable bounds.

Abstract

Very recently, Khoury and Schild [FOCS 2025] showed that any randomized LOCAL algorithm that solves maximal matching requires rounds, where is the number of nodes in the graph and is the maximum degree. This result is shown through a new technique, called round elimination via self-reduction. The lower bound proof is beautiful and presents very nice ideas. However, it spans more than 25 pages of technical details, and hence it is hard to digest and generalize to other problems. Historically, the simplification of proofs and techniques has marked an important turning point in our understanding of the complexity of graph problems. Our paper makes a step forward towards this direction, and provides the following contributions. 1. We present a short and simplified version of the round elimination via self-reduction technique. The simplification of this technique enables us to obtain the following two hardness results. 2. We show that any randomized LOCAL algorithm that solves the maximal -matching problem requires and rounds. We recall that the -matching problem is a generalization of the matching problem where each vertex can have up to incident edges in the matching. As a corollary, for , we obtain a short proof for the maximal matching lower bound shown by Khoury and Schild. 3. Finally, we show that any randomized LOCAL algorithm that properly colors the edges of a graph with colors requires and rounds, for any and any constant .
Paper Structure (47 sections, 17 theorems, 29 equations, 1 table)

This paper contains 47 sections, 17 theorems, 29 equations, 1 table.

Table of Contents

  1. Introduction
  2. Lower bound techniques in the LOCAL model
  3. Round elimination.
  4. Indistinguishability.
  5. Simplifying proofs and techniques marked a turning point
  6. Automatic round elimination.
  7. A breezing proof for KMW.
  8. Round elimination via self-reduction.
  9. Our Contributions
  10. A simplified round elimination via self-reduction technique.
  11. A lower bound for maximal $b$-matching.
  12. A lower bound for $(\Delta + k)$-edge coloring.
  13. Preliminaries
  14. Graphs.
  15. Half-edges and labelings.
  16. ...and 32 more sections

Key Result

Lemma 2.3

There exist two positive constants $\rho, \varepsilon$ such that the following holds. Let $n,\Delta$ be two positive integers such that $n \Delta$ is even, and such that $2 \le \Delta \le n$. There exists a graph $G_{n,\Delta}$ of $n$ nodes with the following properties:

Theorems & Definitions (32)

  • Definition 2.1: View of a node
  • Definition 2.2: Maximal $b$-matching
  • Lemma 2.3: Lemma 2.1 of balliu-boudier-etal-2024-tight-lower-bounds-in-the
  • Definition 3.1: $b$-grabbing
  • Definition 3.2: Saturated node
  • Definition 3.3: Quality and badness
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • proof
  • ...and 22 more