  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.
