Locally Checkable Labelings with Small Messages

TL;DR

This paper investigates the complexity of locally checkable labelings (LCLs) under bandwidth restrictions, revealing that on trees, CONGEST and LOCAL complexities are asymptotically equivalent, but this does not hold for general graphs.

Contribution

It establishes the equivalence of CONGEST and LOCAL complexities for LCLs on trees and provides a counterexample for general graphs showing their differences.

Findings

On trees, CONGEST complexity matches LOCAL complexity asymptotically.

For general graphs, some LCLs require significantly more rounds in CONGEST than in LOCAL.

An explicit LCL problem demonstrates the disparity between CONGEST and LOCAL complexities on general graphs.

Abstract

A rich line of work has been addressing the computational complexity of locally checkable labelings (LCLs), illustrating the landscape of possible complexities. In this paper, we study the landscape of LCL complexities under bandwidth restrictions. Our main results are twofold. First, we show that on trees, the CONGEST complexity of an LCL problem is asymptotically equal to its complexity in the LOCAL model. An analog statement for general (non-LCL) problems is known to be false. Second, we show that for general graphs this equivalence does not hold, by providing an LCL problem for which we show that it can be solved in $O(\log n)$ rounds in the LOCAL model, but requires $\tilde{\Omega}(n^{1/2})$ rounds in the CONGEST model.