Skip to main content

Advertisement

Springer Nature Link
Log in

The mixed center location problem

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

Abstract

This paper studies a new version of the location problem called the mixed center location problem. Let P be a set of n points in the plane. We first consider the mixed 2-center problem, where one of the centers must be in P, and we solve it in \(O(n^2\log n)\) time. Second, we consider the mixed k-center problem, where m of the centers are in P, and we solve it in \(O(n^{m+O(\sqrt{k-m})})\) time. Motivated by two practical constraints, we propose two variations of the problem. Third, we present a 2-approximation algorithm and three heuristics solving the mixed k-center problem (\(k>2\)).

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

References

  • Agarwal PK, Sharir M, Welzl E (2000) The discrete 2-center problem. Discrete Comput Geom 20:287–305

    Article  MathSciNet  Google Scholar 

  • Aurenhammer F, Klein R, Lee DT (2013) Voronoi diagrams and Delaunay triangulations. World Scientific Publishing Company, Singapore

    Book  Google Scholar 

  • Chan TM (1999) More planar two-center algorithms. Comput Geom Theory Appl 13:189–198

    Article  MathSciNet  Google Scholar 

  • Chen D, Chen R (2009) New relaxation-based algorithms for the optimal solution of the continuous and discrete p-center problems. Comput Oper Res 36:1646–1655

    Article  MathSciNet  Google Scholar 

  • Daskin MS (2000) A new approach to solving the vertex p-center problem to optimality: algorithm and computational results. Commun Oper Res Soc Jpn 45(9):428–36

    Google Scholar 

  • Drezener Z (1984) The p-center problem-heuristics and optimal algorithms. J Oper Res Soc 35:741–748

    Google Scholar 

  • Elloumi S, Labb M, Pochet Y (2004) A new formulation and resolution method for the p-center problem. IINFORMS J Comput 16:84–94

    Article  MathSciNet  Google Scholar 

  • Eppstein D (1997) Faster construction of planar two-centers. In: Proceedings of the 8th ACM-SIAM symposium on discrete algorithms, pp 131–138

  • Feder T, Greene D (1988) Optimal algorithms for approximate clustering. In: Proceedings of the 20th ACM symposium on theory of computing, pp 434–444

  • Franti P et al (2015) Clustering datasets. http://cs.uef.fi/sipu/datasets/

  • Gonzalez TF (1985) Clustering to minimize the maximum intercluster distance. Theor Comput Sci 38:293–306

    Article  MathSciNet  Google Scholar 

  • Hershberger J (1993) A faster algorithm for the two-center decision problem. Inf Process Lett 47:23–29

    Article  MathSciNet  Google Scholar 

  • Hershberger J, Suri S (1991) Finding tailored partitions. J Algorithms 12:431–463

    Article  MathSciNet  Google Scholar 

  • Hochbaum DS, Shmoys DB (1985) A best possible heuristic for the k-center problem. Math Oper Res 10(2):180–184

    Article  MathSciNet  Google Scholar 

  • Hwang RZ, Lee RCT, Chang RC (1993) The slab dividing approach to solve the Euclidean P-center problem. Algorithmica 9:1–22

    Article  MathSciNet  Google Scholar 

  • Ilhan T, Ozsoy FA, Pinar MC (2002) An efficient exact algorithm for the vertex p-center problem and computational experiments for different set covering subproblems. http://www.optimization-online.org/DB_HTML/2002/12/588.html

  • Jaromczyk J, Kowaluk M (1994) An efficient algorithm for the Euclidean two-center problem. In: Proceedings 10th ACM symposium on computational geometry, pp 303–311

  • Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems, Part I. The p-centers. SIAM J Appl Math 37:513–538

    Article  MathSciNet  Google Scholar 

  • Megiddo N (1983) Linear-time algorithms for the linear programming in \(R^3\) and related problems. SIAM J Comput 12:759–776

    Article  MathSciNet  Google Scholar 

  • Megiddo N, Supowit K (1984) On the complexity of some common geometric location problems. SIAM J Comput 13:1182–1196

    Article  MathSciNet  Google Scholar 

  • Nagarajan V, Schieber B, Shachnai H (2013) The Eucildean k-supplier problem. In: Goemans M, Correa J (eds) IPCO 2013. LNCS, vol 7801. Springer, Heidelberg, pp 290–301

  • Nenad M, Martine L, Pierre H (2003) Solving the p-center problem with tabu search and variable neighborhood search. Networks 42:48–64

    Article  MathSciNet  Google Scholar 

  • Shamos M, Michael I, Hoey D (1975) Closest-point problems. In: 16th annual symposium on IEEE foundations of computer science, pp 151–162

  • Sharir M (1997) A near-linear algorithm for the planar 2-center problem. Discrete Comput Geom 18:125–134

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yi Xu & Jigen Peng

  2. Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing, 100048, People’s Republic of China

    Jigen Peng

  3. School of Management, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yinfeng Xu

  4. The State Key Lab for Manufacturing Systems Engineering, Xi’an, 710049, People’s Republic of China

    Yinfeng Xu

Authors
  1. Yi Xu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jigen Peng
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Yinfeng Xu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Yi Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, Y., Peng, J. & Xu, Y. The mixed center location problem. J Comb Optim 36, 1128–1144 (2018). https://doi.org/10.1007/s10878-017-0183-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-017-0183-4

Keywords