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A Simple Analysis of Ranking in General Graphs

Mahsa Derakhshan, Mohammad Roghani, Mohammad Saneian, Tao Yu

TL;DR

The paper analyzes Ranking for general graphs, a vertex-iterative randomized greedy algorithm, to determine how closely it can approximate a maximum matching. It introduces a simple combinatorial framework based on augmenting paths and k-wasteful independent sets (k-WIS), using a double-counting argument to relate lower and upper bounds on the number of short augmenting structures. The authors show that if Ranking performed worse than $1/2+c$, with $c\le 0.005$, the resulting counting bounds would contradict each other, yielding a guaranteed approximation of at least $0.505$. This work provides a simpler, LP-free proof of a better-than-0.5 guarantee for Ranking on general graphs and advances the understanding of vertex-iterative randomized greedy methods.

Abstract

We provide a simple combinatorial analysis of the Ranking algorithm, originally introduced in the seminal work by Karp, Vazirani, and Vazirani [KVV90], demonstrating that it achieves a $(1/2 + c)$-approximate matching for general graphs for $c \geq 0.005$.

A Simple Analysis of Ranking in General Graphs

TL;DR

The paper analyzes Ranking for general graphs, a vertex-iterative randomized greedy algorithm, to determine how closely it can approximate a maximum matching. It introduces a simple combinatorial framework based on augmenting paths and k-wasteful independent sets (k-WIS), using a double-counting argument to relate lower and upper bounds on the number of short augmenting structures. The authors show that if Ranking performed worse than , with , the resulting counting bounds would contradict each other, yielding a guaranteed approximation of at least . This work provides a simpler, LP-free proof of a better-than-0.5 guarantee for Ranking on general graphs and advances the understanding of vertex-iterative randomized greedy methods.

Abstract

We provide a simple combinatorial analysis of the Ranking algorithm, originally introduced in the seminal work by Karp, Vazirani, and Vazirani [KVV90], demonstrating that it achieves a -approximate matching for general graphs for .

Paper Structure

This paper contains 9 sections, 6 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Techniques.
  3. Further Related Work.
  4. Preliminaries
  5. Notation and Definitions.
  6. The Ranking Algorithm.
  7. Augmenting Paths.
  8. Combinatorial Tools.
  9. The Analysis

Key Result

Proposition 1.1

Let $\alpha \geq 0$ and $M$ be a maximal matching of $G$ such that $|M| \leq (1/2 + \alpha) \cdot \mu(G)$. Then, at least $(1/2 - 3\alpha)\cdot \mu(G)$ edges of $M$ are in disjoint, length-three augmenting paths.

Figures (1)

  • Figure 1: The red edges are from matching $R$ outputted by Ranking and the blue ones are from OPT. The lower vertices form a 5-WIS as they are the endpoints of five length-three augmenting.

Theorems & Definitions (14)

  • Proposition 1.1: KonradMM12
  • Proposition 1.2: marshall1979inequalities
  • Proposition 1.3: thomas2006elements
  • Definition 2.1: $k$-wasteful independent set ($k$-WIS)
  • Lemma 2.2: Lower Bound on the Expected Number of $k$-WIS
  • proof
  • Claim 2.3
  • proof
  • Claim 2.4
  • proof
  • ...and 4 more