A Local Search Algorithm for the Radius-Constrained k-Median Problem
Abstract
Given a radius and a set X of n points distributed within a metric space, we consider the radius-constrained k-median problem, which combines both the k-center and k-median clustering problems. In this problem, the objective is the same as that of the k-median problem, with the additional constraint that every point x in X must be assigned to a center within the given radius R. This paper proposes an approximation algorithm that achieves a bicriteria approximation ratio of by incorporating local search with a key ball structure. The algorithm constructs a keyball center set to ensure coverage of the points and iteratively refines the solution through subset swaps while satisfying feasibility conditions. Thus, this process maintains coverage while reducing costs. Compared to the state-of-the-art approximation ratio of (8, 4) based on a linear programming formulation for this problem, our approach improves the k-median ratio from 8 to , at the cost of increasing the radius ratio from 4 to 7.
References
[1]
Gramm J, Guo J, Hüffner F, and Niedermeier R Graph-modeled data clustering: exact algorithms for clique generation Theory Comput Syst 2005 38 373-392
[2]
Cormode, G., McGregor, A.: Approximation algorithms for clustering uncertain data. In: Proceedings of the Twenty-seventh ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pp. 191–200 (2008)
[3]
Lin G, Nagarajan C, Rajaraman R, and Williamson DP A general approach for incremental approximation and hierarchical clustering SIAM J. Comput. 2010 39 8 3633-3669
[4]
Kanungo T, Mount DM, Netanyahu NS, Piatko CD, Silverman R, and Wu AY A local search approximation algorithm for -means clustering Comput. Geom. 2004 28 2–3 89-112
[5]
Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristic for -median and facility location problems. In: Proceedings of The Thirty-third Annual ACM Symposium on Theory of Computing (STOC), pp. 21–29 (2001)
[6]
Hochbaum DS and Shmoys DB A best possible heuristic for the -center problem Math. Oper. Res. 1985 10 2 180-184
[7]
Duong K-C, Vrain C, et al. Constrained clustering by constraint programming Artif. Intell. 2017 244 70-94
[8]
Li S On uniform capacitated -median beyond the natural lp relaxation ACM Transactions on Algorithms (TALG) 2017 13 2 1-18
[9]
Bonchi F, Gionis A, Gullo F, Tsourakakis CE, and Ukkonen A Chromatic correlation clustering ACM Transactions on Knowledge Discovery from Data (TKDD) 2015 9 4 1-24
[10]
Cohen-Addad, V., Gupta, A., Hu, L., Oh, H., Saulpic, D.: An improved local search algorithm for -median. In: Proceedings of The 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1556–1612 (2022)
[11]
Li S and Svensson O Approximating -median via pseudo-approximation SIAM J. Comput. 2016 45 2 530-547
[12]
Byrka J, Pensyl T, Rybicki B, Srinivasan A, and Trinh K An improved approximation for -median and positive correlation in budgeted optimization ACM Transactions on Algorithms (TALG) 2017 13 2 1-31
[13]
Jain K, Mahdian M, Markakis E, Saberi A, and Vazirani VV Greedy facility location algorithms analyzed using dual fitting with factor-revealing lp Journal of the ACM (JACM) 2003 50 6 795-824
[14]
Gonzalez TF Clustering to minimize the maximum intercluster distance Theoret. Comput. Sci. 1985 38 293-306
[15]
Hsu W-L and Nemhauser GL Easy and hard bottleneck location problems Discret. Appl. Math. 1979 1 3 209-215
[16]
Jung, C., Kannan, S., Lutz, N.: Service in your neighborhood: Fairness in center location. Foundations of Responsible Computing (FORC) (2020)
[17]
Mahabadi, S., Vakilian, A.: Individual fairness for -clustering. In: Proceedings of the 37th International Conference on Machine Learning (ICML), pp. 6586–6596 (2020)
[18]
Negahbani M and Chakrabarty D Better algorithms for individually fair -clustering Advances in Neural Information Processing Systems (NeurIPS) 2021 34 13340-13351
[19]
Vakilian, A., Yalciner, M.: Improved approximation algorithms for individually fair clustering. In: International Conference on Artificial Intelligence and Statistics, PMLR, pp. 8758–8779 (2022)
[20]
Swamy C Improved approximation algorithms for matroid and knapsack median problems and applications ACM Transactions on Algorithms (TALG) 2016 12 4 1-22
[21]
Alamdari, S., Shmoys, D.: A bicriteria approximation algorithm for the -center and -median problems. In: International Workshop on Approximation and Online Algorithms, Springer, pp. 66–75 (2018)
[22]
Aouad A and Segev D The ordered -median problem: surrogate models and approximation algorithms Math. Program. 2019 177 1–2 55-83
[23]
Byrka, J., Sornat, K., Spoerhase, J.: Constant-factor approximation for ordered -median. In: Proceedings of The 50th Annual ACM SIGACT Symposium on Theory of Computing, pp. 620–631 (2018)
[24]
Nickel, S., Albandoz, J.P.: Location theory. Springer Books (2005)
[25]
Kamiyama N The distance-constrained matroid median problem Algorithmica 2020 82 7 2087-2106
[26]
Chan, T.-H.H., Dinitz, M., Gupta, A.: Spanners with slack. In: European Symposium on Algorithms, pp. 196–207 (2006)
[27]
Charikar M, Makarychev K, and Makarychev Y Local global tradeoffs in metric embeddings SIAM J. Comput. 2010 39 6 2487-2512
Index Terms
- A Local Search Algorithm for the Radius-Constrained k-Median Problem
Index terms have been assigned to the content through auto-classification.
Recommendations
A Local Search Algorithm for Radius-Constrained k-Median
Theory and Applications of Models of ComputationAbstractGiven a set X of n points in a metric space and a radius R, we consider the combining version of k-center and k-median which is with the same objective as k-median, but with the constraint that any x in X is assigned to a center within the range R...
Local Search Algorithms for k-Median and k-Facility Location Problems with Linear Penalties
COCOA 2015: Proceedings of the 9th International Conference on Combinatorial Optimization and Applications - Volume 9486We present two local search algorithms for the k-median and k-facility location problems with linear penalties k-MLP and k-FLPLP, two extensions of the classical k-median and k-facility location problems respectively. We show that the approximation ...
Local search approximation algorithms for the k-means problem with penalties
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?Rd, a penalty ...
Comments
Information & Contributors
Information
Published In
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 30 January 2025
Accepted: 25 December 2024
Author Tags
Qualifiers
- Research-article
Funding Sources
- National Natural Science Foundation of China
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 18 Feb 2025