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### Correlation Clustering with Asymmetric Classification Errors [article]

Jafar Jafarov, Sanchit Kalhan, Konstantin Makarychev, Yury Makarychev
<span title="2021-08-11">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In the Correlation Clustering problem, we are given a weighted graph G with its edges labeled as "similar" or "dissimilar" by a binary classifier. The goal is to produce a clustering that minimizes the weight of "disagreements": the sum of the weights of "similar" edges across clusters and "dissimilar" edges within clusters. We study the correlation clustering problem under the following assumption: Every "similar" edge e has weight 𝐰_e∈[α𝐰, 𝐰] and every "dissimilar" edge e has weight 𝐰_e≥α𝐰
more &raquo; ... ere α≤ 1 and 𝐰>0 is a scaling parameter). We give a (3 + 2 log_e (1/α)) approximation algorithm for this problem. This assumption captures well the scenario when classification errors are asymmetric. Additionally, we show an asymptotically matching Linear Programming integrality gap of Ω(log 1/α).
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2108.05696v1">arXiv:2108.05696v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/usup5vqygzcgxihk67wc6dknfe">fatcat:usup5vqygzcgxihk67wc6dknfe</a> </span>
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### Local Correlation Clustering with Asymmetric Classification Errors [article]

Jafar Jafarov, Sanchit Kalhan, Konstantin Makarychev, Yury Makarychev
<span title="2021-08-11">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We note that our relaxation is simpler than the relaxation used in Kalhan et al. (2019) .  ...  Kalhan et al. (2019) gave an O(n 1 2 − 1 2p · log 1 2 + 1 2p n)-approximation algorithm for minimizing the ℓ p norm of the disagreements vector.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2108.05697v1">arXiv:2108.05697v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/atq4xysqzrdnnl2rxqokk7vuqm">fatcat:atq4xysqzrdnnl2rxqokk7vuqm</a> </span>
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### Improved algorithms for Correlation Clustering with local objectives [article]

Sanchit Kalhan, Konstantin Makarychev, Timothy Zhou
<span title="2019-06-21">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Correlation Clustering is a powerful graph partitioning model that aims to cluster items based on the notion of similarity between items. An instance of the Correlation Clustering problem consists of a graph G (not necessarily complete) whose edges are labeled by a binary classifier as "similar" and "dissimilar". An objective which has received a lot of attention in literature is that of minimizing the number of disagreements: an edge is in disagreement if it is a "similar" edge and is present
more &raquo; ... cross clusters or if it is a "dissimilar" edge and is present within a cluster. Define the disagreements vector to be an n dimensional vector indexed by the vertices, where the v-th index is the number of disagreements at vertex v. Recently, Puleo and Milenkovic (ICML '16) initiated the study of the Correlation Clustering framework in which the objectives were more general functions of the disagreements vector. In this paper, we study algorithms for minimizing ℓ_q norms (q ≥ 1) of the disagreements vector for both arbitrary and complete graphs. We present the first known algorithm for minimizing the ℓ_q norm of the disagreements vector on arbitrary graphs and also provide an improved algorithm for minimizing the ℓ_q norm (q ≥ 1) of the disagreements vector on complete graphs. We also study an alternate cluster-wise local objective introduced by Ahmadi, Khuller and Saha (IPCO '19), which aims to minimize the maximum number of disagreements associated with a cluster. We also present an improved (2 + ε) approximation algorithm for this objective. Finally, we compliment our algorithmic results for minimizing the ℓ_q norm of the disagreements vector with some hardness results.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1902.10829v2">arXiv:1902.10829v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/alrq3xjqbbcevdf425wjbs47bq">fatcat:alrq3xjqbbcevdf425wjbs47bq</a> </span>
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