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A Simple Average-case Analysis of Recursive Randomized Greedy MIS

Mina Dalirrooyfard, Konstantin Makarychev, Slobodan Mitrović

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.

A Simple Average-case Analysis of Recursive Randomized Greedy MIS

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.

Paper Structure

This paper contains 8 sections, 3 theorems, 12 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Counting Recursive Calls using Query Paths
  4. Bounding the Number of Query Paths
  5. Proof of \ref{['item:query-paths-come-from-dangerous']}.
  6. Proof of \ref{['item:new-dangerous-path-had-dangerous-prefix-a-moment-ago']}.
  7. The analysis of $|\mathcal{Q}_{t+1}(a, b)| - |\mathcal{Q}_{t}(a, b)|$, given ${\cal F}_t$.
  8. The analysis of $|\mathcal{D}_{t+1}(a, b)| - |\mathcal{D}_{t}(a, b)|$, given ${\cal F}_t$.

Key Result

Theorem 1.1

Let $G = (V, E)$ be a graph and let $\pi : V \to \{1, \ldots, n\}$ be a uniformly random permutation of its vertices. Then, across all invocations $\texttt{RGMIS}\xspace(u, \pi)$ for $u \in V$, the following holds: for every ordered pair $(a, b)$ with $(a,b) \in E$, the expected number of times $\te

Figures (1)

  • Figure 1: Illustration of a dangerous path $R=(u_1,\ldots,u_k,z)$ and its possible extensions to vertices $w \in A_R$. The ranks of the vertices $u_k$, $z$, and $w_i$ have not yet been revealed by time $t$. Hence $\pi(u_k), \pi(z), \pi(w_i) > t$, and each of these vertices receives rank $t+1$ with probability exactly $\tfrac{1}{n-t}$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 3.1: Query path
  • Remark 3.2
  • Claim 3.3
  • proof
  • Definition 3.4: Dangerous path
  • Claim 3.5: Evolution of query and dangerous paths
  • proof
  • Claim 3.6
  • ...and 3 more