Variants of local algorithms on sparse graphs

TL;DR

This paper demonstrates that for constructing structures like near-maximum matchings on bounded-degree graphs, local algorithms can achieve optimal results without preprocessing by using global randomization instead.

Contribution

It shows that preprocessing in local algorithms is unnecessary, as similar performance can be obtained through global randomization without prior graph analysis.

Findings

Preprocessing can be replaced by global randomization in local algorithms.

Local algorithms without preprocessing can match the performance of those with preprocessing.

The approach simplifies the design of local algorithms for sparse graphs.

Abstract

Suppose we want to construct some structure on a bounded-degree graph, e.g., an almost maximum matching, and we want to decide about each edge depending only on its constant-radius neighborhood. We examine and compare the strengths of different extensions of these local algorithms. A common extension is to use preprocessing, which means that we can make some calculation about the whole graph, and each local decision can also depend on this calculation. In this paper, we show that preprocessing is needless: if a nearly optimal local algorithm uses preprocessing, then the same can be achieved by a local algorithm without preprocessing, but with a global randomization.