Simultaneously Approximating All Norms for Massively Parallel Correlation Clustering

TL;DR

This paper introduces an efficient algorithm for correlation clustering that approximates all monotone symmetric norms simultaneously, significantly improving the approximation ratio and enabling scalable parallel computation.

Contribution

The authors develop a novel approach to approximate all monotone symmetric norms in correlation clustering with a unified algorithm, improving the approximation ratio from 6348 to 63.3.

Findings

Achieved a 63.3-approximation for all monotone symmetric norms.

Reduced the problem to approximating all top-k norms simultaneously.

Algorithm can run in nearly linear time and in MPC model with poly-logarithmic rounds.

Abstract

We revisit the simultaneous approximation model for the correlation clustering problem introduced by Davies, Moseley, and Newman[DMN24]. The objective is to find a clustering that minimizes given norms of the disagreement vector over all vertices. We present an efficient algorithm that produces a clustering that is simultaneously a $63.3$-approximation for all monotone symmetric norms. This significantly improves upon the previous approximation ratio of $6348$ due to Davies, Moseley, and Newman[DMN24], which works only for $\ell_p$-norms. To achieve this result, we first reduce the problem to approximating all top-$k$ norms simultaneously, using the connection between monotone symmetric norms and top-$k$ norms established by Chakrabarty and Swamy [CS19]. Then we develop a novel procedure that constructs a $12.66$-approximate fractional clustering for all top-$k$ norms. Our $63.3$-approximation ratio is obtained by combining this with the $5$-approximate rounding algorithm by Kalhan, Makarychev, and Zhou[KMZ19]. We then demonstrate that with a loss of $\epsilon$ in the approximation ratio, the algorithm can be adapted to run in nearly linear time and in the MPC (massively parallel computation) model with poly-logarithmic number of rounds. By allowing a further trade-off in the approximation ratio to $(359+\epsilon)$, the number of MPC rounds can be reduced to a constant.