In the k-means problem with penalties, we are given a data set D subset of R-l of n points where each point j is an element of D is associated with a penalty cost p(j) and an integer k. The goal is to choose a set CS ...
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In the k-means problem with penalties, we are given a data set D subset of R-l of n points where each point j is an element of D is associated with a penalty cost p(j) and an integer k. The goal is to choose a set CS subset of R-l with vertical bar CS vertical bar <= k and a penalized subset D-p subset of D to minimize the sum of the total squared distance from the points in D\D-p to CS and the total penalty cost of points in D-p, namely Sigma(j is an element of D\Dp) d(2)(j, CS)+ Sigma(j is an element of Dp) p(j). We employ the primaldual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ epsilon) for the k-means problem with penalties, improving the previous best approximation ratio 19.849 + epsilon for this problem given by Feng et al. in Proceedings of FAW(2019).
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