Core Concepts
The authors propose the cluster LP as a strong linear program that unifies previous relaxations for the Correlation Clustering problem, and present improved approximation algorithms and hardness results based on this new framework.
Abstract
The content discusses the Correlation Clustering problem, where the goal is to find a partition of vertices in a graph that minimizes the sum of positive edges between different clusters and negative edges within the same cluster.
The key highlights are:
The authors propose the cluster LP as a strong linear program that captures all previous relaxations for the Correlation Clustering problem. They show that the cluster LP can be approximately solved in polynomial time.
Using the cluster LP framework, the authors present a simple rounding algorithm and provide two analyses - one analytically proving a 1.49-approximation and another using a factor-revealing SDP to show a 1.437-approximation. These results significantly improve upon the previous best 1.73-approximation.
The authors also prove an integrality gap of 4/3 for the cluster LP, showing that their 1.437-upper bound cannot be drastically improved. This gap instance directly inspires an improved NP-hardness of approximation with a ratio of 24/23 ≈ 1.042, which was not known before.
The authors introduce new techniques, such as a simpler and better preclustering procedure and principled methods for analyzing the performance of the rounding algorithms, which lead to the improved approximation guarantees.