Growing a Random Maximal Independent Set Produces a 2-approximate Vertex Cover

TL;DR

This paper introduces a fast, simple 2-approximation algorithm for minimum weighted vertex cover, extending known greedy algorithms and connecting them to correlation clustering for broader applications.

Contribution

It provides a new analysis and generalization of a greedy maximal independent set algorithm for weighted graphs, linking it to correlation clustering and related problems.

Findings

Proves the algorithm achieves a 2-approximation for weighted vertex cover.

Establishes a connection between independent set algorithms and correlation clustering.

Develops fast algorithms for edge-deletion problems reducible to vertex cover.

Abstract

This paper presents a fast and simple new 2-approximation algorithm for minimum weighted vertex cover. The unweighted version of this algorithm is equivalent to a well-known greedy maximal independent set algorithm. We prove that this independent set algorithm produces a 2-approximate vertex cover, and we provide a principled new way to generalize it to node-weighted graphs. Our analysis is inspired by connections to a clustering objective called correlation clustering. To demonstrate the relationship between these problems, we show how a simple Pivot algorithm for correlation clustering implicitly approximates a special type of hypergraph vertex cover problem. Finally, we use implicit implementations of this maximal independent set algorithm to develop fast and simple 2-approximation algorithms for certain edge-deletion problems that can be reduced to vertex cover in an approximation preserving way.