A Simple Average-case Analysis of Recursive Randomized Greedy MIS

TL;DR

This paper presents a simpler, more transparent average-case analysis of the recursive randomized greedy algorithm for computing a maximal independent set, matching previous bounds.

Contribution

It introduces a new, simpler analysis method inspired by recent work on correlation clustering, improving understanding of the MIS algorithm's complexity.

Findings

The new analysis achieves the same average recursive call bound as previous work.

It simplifies the proof of the expected complexity of the recursive randomized greedy MIS algorithm.

Abstract

We revisit the complexity analysis of the recursive version of the randomized greedy algorithm for computing a maximal independent set (MIS), originally analyzed by Yoshida, Yamamoto, and Ito (2009). They showed that, on average per vertex, the expected number of recursive calls made by this algorithm is upper bounded by the average degree of the input graph. While their analysis is clever and intricate, we provide a significantly simpler alternative that achieves the same guarantee. Our analysis is inspired by the recent work of Dalirrooyfard, Makarychev, and Mitrovi\'c (2024), who developed a potential-function-based argument to analyze a new algorithm for correlation clustering. We adapt this approach to the MIS setting, yielding a more direct and arguably more transparent analysis of the recursive randomized greedy MIS algorithm.