Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

On parameterized approximation algorithms for balanced clustering

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

Balanced clustering is a frequently encountered problem in applications requiring balanced class distributions, which generalizes the standard clustering problem in that the number of clients connected to each facility is constrained by the given lower and upper bounds. It was known that both the problems of balanced k-means and k-median are W[2]-hard if parameterized by k, implying that the existences of FPT(k)-time exact algorithms for these problems are unlikely. In this paper, we give FPT(k)-time \((9+\epsilon )\)-approximation and \((3+\epsilon )\)-approximation algorithms for balanced k-means and k-median respectively, improving upon the previous best approximation ratios of \(86.9+\epsilon \) and \(7.2+\epsilon \) obtained in the same time. Our main technical contribution and the crucial step in getting the improved ratios is a different random sampling method for selecting opened facilities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availibility

Enquiries about data availability should be directed to the authors.

References

  • Adamczyk M, Byrka J, Marcinkowski J, Meesum SM, Wlodarczyk M (2019) Constant-factor FPT approximation for capacitated \(k\)-median. In: Proceedings of of 27th annual European symposium on algorithms (ESA), pp 1:1–1:14

  • Ahmadian S, Norouzi-Fard A, Svensson O, Ward J (2020) Better guarantees for \(k\)-means and Euclidean \(k\)-median by primal-dual algorithms. SIAM J Comput 49(4)

  • Aydin K, Bateni M, Mirrokni VS (2016) Distributed balanced partitioning via linear embedding. In: Proceedings of 9th ACM international conference on web search and data mining (WSDM), pp 387–396

  • Bandyapadhyay S, Fomin FV, Simonov K (2021) On coresets for fair clustering in metric and Euclidean spaces and their applications. In: Proceedings of 48th international colloquium on automata, languages, and programming (ICALP), pp 23:1–23:15

  • Bateni M, Bhaskara A, Lattanzi S, Mirrokni VS (2014) Distributed balanced clustering via mapping coresets. In: Proceedings of 27th annual conference on neural information processing systems (NeurIPS), pp 2591–2599

  • Borgwardt S, Brieden A, Gritzmann P (2017) An LP-based \(k\)-means algorithm for balancing weighted point sets. Eur J Oper Res 263(2):349–355

    Article  MathSciNet  MATH  Google Scholar 

  • Byrka J, Pensyl T, Rybicki B, Srinivasan A, Trinh K (2017) An improved approximation for \(k\)-median and positive correlation in budgeted optimization. ACM Trans Algorithms 13(2):23:1-23:31

    Article  MathSciNet  MATH  Google Scholar 

  • Chen K (2009) On coresets for \(k\)-median and \(k\)-means clustering in metric and Euclidean spaces and their applications. SIAM J Comput 39(3):923–947

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen-Addad V, Esfandiari H, Mirrokni VS, Narayanan S (2022) Improved approximations for Euclidean \(k\)-means and \(k\)-median, via nested quasi-independent sets. In: Proceedings of 54th annual ACM SIGACT symposium on theory of computing (STOC). ACM, pp 1621–1628

  • Cohen-Addad V, Gupta A, Kumar A, Lee E, Li J (2019) Tight FPT approximations for \(k\)-median and \(k\)-means. In: Proceedings of 46th international colloquium on automata, languages, and programming (ICALP), pp 42:1–42:14

  • Cohen-Addad V, Li J (2019) On the fixed-parameter tractability of capacitated clustering. In: Proceedings of 46th international colloquium on automata, languages, and programming (ICALP), pp 41:1–41:14

  • Demirci HG, Li S (2016) Constant approximation for capacitated \(k\)-median with \((1+\epsilon )\)-capacity violation. In: Proceedings of 43rd international colloquium on automata, languages, and programming (ICALP), pp 73:1–73:14

  • Dick T, Li M, Pillutla VK, White C, Balcan N, Smola AJ (2017) Data driven resource allocation for distributed learning. In: Proceedings of 20th international conference on artificial intelligence and statistics (AISTATS), pp 662–671

  • Ding H (2020) Faster balanced clusterings in high dimension. Theor Comput Sci 842:28–40

    Article  MathSciNet  MATH  Google Scholar 

  • Feng Q, Zhang Z, Huang Z, Xu J, Wang J (2020) A unified framework of FPT approximation algorithms for clustering problems. In: Proceedings of 31st international symposium on algorithms and computation (ISAAC), pp 5:1–5:17

  • Guha S, Khuller S (1999) Greedy strikes back: improved facility location algorithms. J Algorithms 31(1):228–248

    Article  MathSciNet  MATH  Google Scholar 

  • Han L, Hao C, Wu C, Zhang Z (2020) Approximation algorithms for the lower-bounded \(k\)-median and its generalizations. In: Proceedings of 26th international conference on computing and combinatorics (COCOON), pp 627–639

  • Li S (2016) Approximating capacitated \(k\)-median with \((1+\epsilon )k\) open facilities. In: Proceedings of 27th annual ACM-SIAM symposium on discrete algorithms (SODA), pp 786–796

  • Li S (2017) On uniform capacitated \(k\)-median beyond the natural LP relaxation. ACM Trans Algorithms 13(2):22:1-22:18

    Article  MathSciNet  MATH  Google Scholar 

  • Li S, Svensson O (2016) Approximating \(k\)-median via pseudo-approximation. SIAM J Comput 45(2):530–547

    Article  MathSciNet  MATH  Google Scholar 

  • Lin W, He Z, Xiao M (2019) Balanced clustering: A uniform model and fast algorithm. In: Proceedings of 28th international joint conference on artificial intelligence (IJCAI), pp 2987–2993

  • Manurangsi P, Raghavendra P (2017) A birthday repetition theorem and complexity of approximating dense CSPs. In: Proceedings of 44th international colloquium on automata, languages, and programming (ICALP), pp 78:1–78:15

  • Qi L, Wang R, Hu C, Li S, He Q, Xu X (2019) Time-aware distributed service recommendation with privacy-preservation. Inf Sci 480:354–364

    Article  Google Scholar 

  • Siavoshi S, Kavian YS, Sharif H (2016) Load-balanced energy efficient clustering protocol for wireless sensor networks. IET Wirel Sens Syst 6(3):67–73

    Article  Google Scholar 

  • Wu X, Shi F, Guo Y, Zhang Z, Huang J, Wang J (2022) An approximation algorithm for lower-bounded \(k\)-median with constant factor. Sci China Inf Sci 65(4):140601:1–140601:9

  • Xu X, Du Z, Chen X, Cai C (2019) Confidence consensus-based model for large-scale group decision making: a novel approach to managing non-cooperative behaviors. Inf Sci 477:410–427

    Article  Google Scholar 

  • Xu Y, Möhring RH, Xu D, Zhang Y, Zou Y (2020) A constant FPT approximation algorithm for hard-capacitated \(k\)-means. Optim Eng 21(3):709–722

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang D, Qiao Y, She L, Shen R, Ren J, Zhang Y (2019) Two time-scale resource management for green internet of things networks. IEEE Internet Things J 6(1):545–556

    Article  Google Scholar 

Download references

Funding

This work was supported by National Natural Science Foundation of China (62202161, 62172446), National Key R &D Program of China (2021YFC3300603), Open Project of Xiangjiang Laboratory (22XJ02002), Central South University Research Programme of Advanced Interdisciplinary Studies (2023QYJC023), and Scientific Research Fund of Hunan Provincial Education Department (20C0538, 21A0376). A preliminary version of this paper has appeared in Proceedings of the 15th International Conference on Combinatorial Optimization and Applications (COCOA), 2021, pp. 629-640.

Author information

Authors and Affiliations

  1. School of Computer Science and Engineering, Central South University, Changsha, 410000, People’s Republic of China

    Xiangyan Kong & Qilong Feng

  2. School of Frontier Crossover Studies, Hunan University of Technology and Business, Changsha, 410000, People’s Republic of China

    Zhen Zhang

Authors
  1. Xiangyan Kong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qilong Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen Zhang.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kong, X., Zhang, Z. & Feng, Q. On parameterized approximation algorithms for balanced clustering. J Comb Optim 45, 49 (2023). https://doi.org/10.1007/s10878-022-00980-w

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10878-022-00980-w

Keywords

Search

Navigation