Local Correlation Clustering with Asymmetric Classification Errors

• URL: http://arxiv.org/abs/2108.05697v1
• Date: Wed, 11 Aug 2021 12:31:48 GMT
• Title: Local Correlation Clustering with Asymmetric Classification Errors
• Authors: Jafar Jafarov, Sanchit Kalhan, Konstantin Makarychev and Yury Makarychev
• Abstract summary: In the Correlation Clustering problem, we are given a complete weighted graph $G$ with its edges labeled as "similar" and "dissimilar" For a clustering $mathcalC$ of graph $G$, a similar edge is in disagreement with $mathcalC$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $mathcalC$ if its endpoints belong to the same cluster. We produce a clustering that minimizes the $ell_p$ norm of the disagreements vector for $pgeq 1$
• Score: 12.277755088736864
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In the Correlation Clustering problem, we are given a complete weighted graph $G$ with its edges labeled as "similar" and "dissimilar" by a noisy binary classifier. For a clustering $\mathcal{C}$ of graph $G$, a similar edge is in disagreement with $\mathcal{C}$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $\mathcal{C}$ if its endpoints belong to the same cluster. The disagreements vector, $\text{dis}$, is a vector indexed by the vertices of $G$ such that the $v$-th coordinate $\text{dis}_v$ equals the weight of all disagreeing edges incident on $v$. The goal is to produce a clustering that minimizes the $\ell_p$ norm of the disagreements vector for $p\geq 1$. We study the $\ell_p$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in the range of $[\alpha\mathbf{w},\mathbf{w}]$ and every dissimilar edge has weight at least $\alpha\mathbf{w}$ (where $\alpha \leq 1$ and $\mathbf{w}>0$ is a scaling parameter). We give an $O\left((\frac{1}{\alpha})^{\frac{1}{2}-\frac{1}{2p}}\cdot \log\frac{1}{\alpha}\right)$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.

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