Local Mending

TL;DR

This paper introduces the concept of mendability in graph problems, linking local repairability to efficient distributed algorithms, and explores the structural landscape of mendability across different graph classes.

Contribution

It formalizes the notion of mendability, connects it to distributed complexity, and characterizes its structure in trees and general graphs.

Findings

O(1)-mendable problems are solvable in O(log* n) rounds.

In paths and cycles, O(log* n)-solvability implies existence of an O(1)-mendable restriction.

In trees, mending radius is either O(1), Θ(log n), or Θ(n).

Abstract

In this work we introduce the graph-theoretic notion of mendability: for each locally checkable graph problem we can define its mending radius, which captures the idea of how far one needs to modify a partial solution in order to "patch a hole." We explore how mendability is connected to the existence of efficient algorithms, especially in distributed, parallel, and fault-tolerant settings. It is easy to see that $O(1)$-mendable problems are also solvable in $O(\log^* n)$ rounds in the LOCAL model of distributed computing. One of the surprises is that in paths and cycles, a converse also holds in the following sense: if a problem $\Pi$ can be solved in $O(\log^* n)$, there is always a restriction $\Pi' \subseteq \Pi$ that is still efficiently solvable but that is also $O(1)$-mendable. We also explore the structure of the landscape of mendability. For example, we show that in trees, the mending radius of any locally checkable problem is $O(1)$, $\Theta(\log n)$, or $\Theta(n)$, while in general graphs the structure is much more diverse.