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Abstract: We consider inapproximability of the correlation clustering problem definedas follows: Given a graph $G = V,E$ where each edge is labeled either -+-similar or - dissimilar, correlation clustering seeks to partition thevertices into clusters so that the number of pairs correctly resp.incorrectly classified with respect to the labels is maximized resp.minimized. The two complementary problems are called MaxAgree and MinDisagree,respectively, and have been studied on complete graphs, where every edge islabeled, and general graphs, where some edge might not have been labeled.Natural edge-weighted versions of both problems have been studied as well. LetS-MaxAgree denote the weighted problem where all weights are taken from set S,we show that S-MaxAgree with weights bounded by $O|V|^{1-2-\delta}$essentially belongs to the same hardness class in the following sense: if thereis a polynomial time algorithm that approximates S-MaxAgree within a factor of$\lambda = O\log{|V|}$ with high probability, then for any choice of S,S-MaxAgree can be approximated in polynomial time within a factor of $\lambda+ \epsilon$, where $\epsilon > 0$ can be arbitrarily small, with highprobability. A similar statement also holds for $S-MinDisagree. This resultimplies it is hard assuming $NP eq RP$ to approximate unweighted MaxAgreewithin a factor of $80-79-\epsilon$, improving upon a previous known factor of$116-115-\epsilon$ by Charikar et. al. \cite{Chari05}.

Autor: Jinsong Tan

Fuente: https://arxiv.org/

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