Local algorithms, regular graphs of large girth, and random regular graphs

TL;DR

This paper develops a framework for analyzing algorithms on large-girth regular graphs, enabling the transfer of results from random regular graphs to deterministic ones, and provides new bounds on various graph parameters.

Contribution

It introduces a general class of algorithms and a method to transfer results from random to deterministic regular graphs with large girth, enabling new bounds on graph properties.

Findings

New bounds on maximum independent sets in large-girth regular graphs

Bounds on minimum dominating sets and other graph parameters

First-time transfer of results from random to deterministic regular graphs

Abstract

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This is an uncommon direction of transfer of results, which is usually from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As examples, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum and minimum bisection, maximum $k$-independent sets, minimum $k$-dominating sets and minimum connected and weakly-connected dominating sets in $r$-regular graphs with large girth.