Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Minimum s-t Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization.

Here, we are given a graph with edges labeled + or - and the goal is to produce a clustering that agrees with the labels as much as possible: + edges within clusters and - edges across clusters.

The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective, e.g., minimizing the total number of disagreements or maximizing the total number of agreements.

We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective.

This naturally gives rise to a family of basic min-max graph cut problems.

A prototypical representative is Min-Max s-t Cut: find an s-t cut minimizing the largest number of cut edges incident on any node.

In this talk we will give a short introduction of Correlation Clustering and discuss the following results:

- an O(\sqrt{n})-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems)
- a remarkably simple 7-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of 48)
- a (1/(2+epsilon))-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the (1/(4+epsilon))-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games.

Joint work with Moses Charikar and Neha Gupta.