A Simple Average-case Analysis of Recursive Randomized Greedy MIS

2026-04-01

AI Summary

This work investigates the average-case time complexity of the recursive randomized greedy algorithm for the Maximal Independent Set (MIS) problem. Introducing, for the first time, a potential function method to analyze this algorithm, the study combines probabilistic and graph-theoretic tools to significantly streamline the existing proof framework. The main contribution lies in providing a more concise and intuitive proof that the expected number of recursive calls per vertex is at most the graph’s average degree. This bound matches the best-known result in the literature, yet the derivation is markedly clearer and more efficient, offering a fresh theoretical perspective on the analysis of MIS algorithms.

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 Mitrović (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.

Problem

• maximal independent set
• randomized greedy algorithm
• average-case analysis
• recursive algorithm
• graph algorithms

Innovation

• maximal independent set
• randomized greedy algorithm
• average-case analysis
• potential function
• recursive algorithm