Simultaneously Approximating All $\ell_p$-norms in Correlation Clustering

TL;DR

This paper introduces a combinatorial algorithm for correlation clustering that approximates all $\, ext{ell}_p$-norms of disagreement simultaneously, achieving the first such approximation with optimal trade-offs and improved runtime.

Contribution

It presents the first combinatorial approximation algorithm for all $\, ext{ell}_p$-norms in correlation clustering, including the $\, ext{ell}_2$-norm, with faster runtime than previous methods.

Findings

First combinatorial algorithm for all $\, ext{ell}_p$-norms in correlation clustering.

Achieves $O(1)$-approximation for all norms simultaneously.

Faster runtime $O(n^)$ and $O(n\u03b4^2 \, ext{log} n)$ under degree constraints.

Abstract

This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously $O(1)$-approximate for all $\ell_p$-norms of the disagreement vector; in other words, a combinatorial $O(1)$-approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the $\ell_2$-norm objective, and more generally the first combinatorial algorithm for the $\ell_p$-norm objective when $1 < p < \infty$. It is also faster than all previous algorithms that minimize the $\ell_p$-norm of the disagreement vector, with run-time $O(n^\omega)$, where $O(n^\omega)$ is the time for matrix multiplication on $n \times n$ matrices. When the maximum positive degree in the graph is at most $\Delta$, this can be improved to a run-time of $O(n\Delta^2 \log n)$.