Local search approximation algorithms for the k-means problem with penalties

Dongmei Zhang (), Chunlin Hao (), Chenchen Wu (), Dachuan Xu () and Zhenning Zhang ()
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Dongmei Zhang: Shandong Jianzhu University
Chunlin Hao: Beijing University of Technology
Chenchen Wu: Tianjin University of Technology
Dachuan Xu: Beijing University of Technology
Zhenning Zhang: Beijing University of Technology

Journal of Combinatorial Optimization, 2019, vol. 37, issue 2, No 3, 439-453

Abstract: Abstract In this paper, we study the k-means problem with (nonuniform) penalties (k-MPWP) which is a natural generalization of the classic k-means problem. In the k-MPWP, we are given an n-client set $$ {\mathcal {D}} \subset {\mathbb {R}}^d$$ D ⊂ R d , a penalty cost $$p_j>0$$ p j > 0 for each $$j \in {\mathcal {D}}$$ j ∈ D , and an integer $$k \le n$$ k ≤ n . The goal is to open a center subset $$F \subset {\mathbb {R}}^d$$ F ⊂ R d with $$ |F| \le k$$ | F | ≤ k and to choose a client subset $$P \subseteq {\mathcal {D}} $$ P ⊆ D as the penalized client set such that the total cost (including the sum of squares of distance for each client in $$ {\mathcal {D}} \backslash P $$ D \ P to the nearest open center and the sum of penalty cost for each client in P) is minimized. We offer a local search $$( 81+ \varepsilon )$$ ( 81 + ε ) -approximation algorithm for the k-MPWP by using single-swap operation. We further improve the above approximation ratio to $$( 25+ \varepsilon )$$ ( 25 + ε ) by using multi-swap operation.

Keywords: Approximation algorithm; k-means; Penalty; Local search (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-018-0278-6

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