An improved primal-dual approximation algorithm for the k-means problem with penalties

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 penal...

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Bibliographic Details
Published in:Mathematical structures in computer science Vol. 32; no. 2; pp. 151 - 163
Main Authors: Ren, Chunying, Xu, Dachuan, Du, Donglei, Li, Min
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.02.2022
Subjects:
Algorithms
Approximation
Artificial intelligence
Computer science
Customers
Fines & penalties
Mathematical analysis
Polynomials
Special Issue: Theory and Applications of Models of Computation (TAMC 2020)
approximation algorithm
linear program
K-means problem with penalties
JV algorithm
ISSN:0960-1295, 1469-8072
Online Access:Get full text
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Summary: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).
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ISSN:0960-1295
1469-8072
DOI:10.1017/S0960129521000104