Clustering What Matters in Constrained Settings Improved Outlier to Outlier-Free Reductions

Constrained clustering problems generalize classical clustering formulations, e.g., k -median , k -means , by imposing additional constraints on the feasibility of a clustering. There has been significant recent progress in obtaining approximation algorithms for these problems, both in the metric an...

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Veröffentlicht in:	Algorithmica Jg. 87; H. 8; S. 1178 - 1198
Hauptverfasser:	Jaiswal, Ragesh, Kumar, Amit
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.08.2025
Schlagworte:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation Outlier Theory of computation Constrained Clustering Facility location and clustering
ISSN:	0178-4617, 1432-0541
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Abstract	Constrained clustering problems generalize classical clustering formulations, e.g., k -median , k -means , by imposing additional constraints on the feasibility of a clustering. There has been significant recent progress in obtaining approximation algorithms for these problems, both in the metric and the Euclidean settings. However, the outlier version of these problems, where the solution is allowed to leave out m points from the clustering, is not well understood. In this work, we give a general framework for reducing the outlier version of a constrained k -median or k -means problem to the corresponding outlier-free version with only ( 1 + ε ) -loss in the approximation ratio. The reduction is obtained by mapping the original instance of the problem to f ( k , m , ε ) instances of the outlier-free version, where f ( k , m , ε ) = k + m ε O ( m ) . As specific applications, we get the following results: First FPT ( in the parameters k and m ) ( 1 + ε ) -approximation algorithm for the outlier version of capacitated k -median and k -means in Euclidean spaces with hard capacities. First FPT ( in the parameters k and m ) ( 3 + ε ) and ( 9 + ε ) approximation algorithms for the outlier version of capacitated k -median and k -means , respectively, in general metric spaces with hard capacities. First FPT ( in the parameters k and m ) ( 2 - δ ) -approximation algorithm for the outlier version of the k -median problem under the Ulam metric. Our work generalizes the results of Bhattacharya et al. and Agrawal et al. to a larger class of constrained clustering problems. Further, our reduction works for arbitrary metric spaces and so can extend clustering algorithms for outlier-free versions in both Euclidean and arbitrary metric spaces.
AbstractList	Constrained clustering problems generalize classical clustering formulations, e.g., k -median , k -means , by imposing additional constraints on the feasibility of a clustering. There has been significant recent progress in obtaining approximation algorithms for these problems, both in the metric and the Euclidean settings. However, the outlier version of these problems, where the solution is allowed to leave out m points from the clustering, is not well understood. In this work, we give a general framework for reducing the outlier version of a constrained k -median or k -means problem to the corresponding outlier-free version with only ( 1 + ε ) -loss in the approximation ratio. The reduction is obtained by mapping the original instance of the problem to f ( k , m , ε ) instances of the outlier-free version, where f ( k , m , ε ) = k + m ε O ( m ) . As specific applications, we get the following results: First FPT ( in the parameters k and m ) ( 1 + ε ) -approximation algorithm for the outlier version of capacitated k -median and k -means in Euclidean spaces with hard capacities. First FPT ( in the parameters k and m ) ( 3 + ε ) and ( 9 + ε ) approximation algorithms for the outlier version of capacitated k -median and k -means , respectively, in general metric spaces with hard capacities. First FPT ( in the parameters k and m ) ( 2 - δ ) -approximation algorithm for the outlier version of the k -median problem under the Ulam metric. Our work generalizes the results of Bhattacharya et al. and Agrawal et al. to a larger class of constrained clustering problems. Further, our reduction works for arbitrary metric spaces and so can extend clustering algorithms for outlier-free versions in both Euclidean and arbitrary metric spaces.
Author	Jaiswal, Ragesh Kumar, Amit
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Keywords	Outlier Theory of computation Constrained Clustering Facility location and clustering
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation
Subtitle	Improved Outlier to Outlier-Free Reductions
Title	Clustering What Matters in Constrained Settings
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