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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 C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector is a vector indexed by the vertices of G such that the v-th coordinate of the disagreements vector equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α ≤ 1 and w > 0 is a scaling parameter). We give an O((1/α)^{1/2−1/(2p)} log 1/α) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.

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