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An improved primal-dual approximation algorithm for the k-means problem with penalties
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- Journal:
- Mathematical Structures in Computer Science / Volume 32 / Issue 2 / February 2022
- Published online by Cambridge University Press:
- 16 August 2021, pp. 151-163
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- Article
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In the k-means problem with penalties, we are given a data set
$${\cal D} \subseteq \mathbb{R}^\ell $$ of n points where each point 
$$j \in {\cal D}$$ is associated with a penalty cost pj and an integer k. The goal is to choose a set 
$${\rm{C}}S \subseteq {{\cal R}^\ell }$$ with |CS| ≤ k and a penalized subset 
$${{\cal D}_p} \subseteq {\cal D}$$ to minimize the sum of the total squared distance from the points in D / Dp to CS and the total penalty cost of points in Dp, namely 
$$\sum\nolimits_{j \in {\cal D}\backslash {{\cal D}_p}} {d^2}(j,{\rm{C}}S) + \sum\nolimits_{j \in {{\cal D}_p}} {p_j}$$. We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).
