Understanding the Cluster LP for Correlation Clustering

TL;DR

This paper introduces the cluster LP for correlation clustering, providing new rounding algorithms and analysis that improve approximation guarantees, and establishes an integrality gap and hardness results.

Contribution

It proposes the cluster LP as a unified framework, offers improved approximation algorithms with rigorous analysis, and proves an integrality gap and hardness bounds.

Findings

Achieves a 1.485-approximation for correlation clustering.

Establishes an integrality gap of 4/3 for the cluster LP.

Provides an NP-hardness of approximation ratio of 24/23.

Abstract

In the classic Correlation Clustering problem introduced by Bansal, Blum, and Chawla (FOCS 2002), the input is a complete graph where edges are labeled either $+$ or $-$, and the goal is to find a partition of the vertices that minimizes the sum of the +edges across parts plus the sum of the -edges within parts. In recent years, Chawla, Makarychev, Schramm and Yaroslavtsev (STOC 2015) gave a 2.06-approximation by providing a near-optimal rounding of the standard LP, and Cohen-Addad, Lee, Li, and Newman (FOCS 2022, 2023) finally bypassed the integrality gap of 2 for this LP giving a $1.73$-approximation for the problem. In order to create a simple and unified framework for Correlation Clustering similar to those for typical approximate optimization tasks, we propose the cluster LP as a strong linear program for Correlation Clustering. We demonstrate the power of the cluster LP by presenting new rounding algorithms, and providing two analyses, one analytically proving a 1.56-approximation and the other solving a factor-revealing SDP to show a 1.485-approximation. Both proofs introduce principled methods by which to analyze the performance of the algorithm, resulting in a significantly improved approximation guarantee. Finally, we prove an integrality gap of $4/3$ for the cluster LP, showing our 1.485-upper bound cannot be drastically improved. Our gap instance directly inspires an improved NP-hardness of approximation with a ratio $24/23 \approx 1.042$; no explicit hardness ratio was known before.