Randomized greedy algorithms for independent sets and matchings in regular graphs: Exact results and finite girth corrections

TL;DR

This paper analyzes the performance of a simple greedy algorithm for finding large independent sets and matchings in regular graphs, providing exact bounds, finite girth corrections, and results for arbitrary graphs and weighted cases.

Contribution

It introduces explicit formulas and finite girth corrections for the greedy algorithm's performance on regular graphs, and extends results to matchings, arbitrary graphs, and weighted cases.

Findings

Expected independent set size in regular graphs with girth g.

Greedy algorithm yields nearly perfect matchings for large r and g.

Concentration of independent set and matching sizes around the mean.

Abstract

We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant degree regular graphs. We show that for $r$-regular graphs with $n$ nodes and girth at least $g$, the algorithm finds an independent set of expected cardinality $f(r)n - O\big(\frac{(r-1)^{\frac{g}{2}}}{\frac{g}{2}!} n\big)$, where $f(r)$ is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degree $r$ and girth $g$ are large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm in \emph{arbitrary} bounded degree graphs is concentrated around the mean. Finally, we analyze the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.