A PTAS for the Geometric Connected Facility Location Problem


We consider the Geometric Connected Facility Location Problem (GCFLP): given a set of clients \(\mathcal {C} \subset \mathbb {R}^{d}\), one wants to select a set of locations \(F \subset \mathbb {R}^{d}\) where to open facilities, each at a fixed cost f≥0. For each client \(j \in \mathcal {C}\), one has to choose to either connect it to an open facility ϕ(j)∈F paying the Euclidean distance between j and ϕ(j), or pay a given penalty cost π(j). The facilities must also be connected by a tree T, whose cost is M (T), where M≥1 and (T) is the total Euclidean length of edges in T. The multiplication by M reflects the fact that interconnecting two facilities is typically more expensive than connecting a client to a facility. The objective is to find a solution with minimum cost. In this paper, we present a Polynomial-Time Approximation Scheme (PTAS) for the two-dimensional GCFLP. Our algorithm also leads to a PTAS for the two-dimensional Geometric Connected k-medians, when f=0, but only k facilities may be opened.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    That is, the vertex \((j,i^{\prime }) \in \mathcal {V}\) that represents the connected component \(i^{\prime }\) of G j is part of connected component i of G.


  1. 1.

    Arora, S.: Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and Other Geometric Problems. J. ACM 45(5), 753–782 (1998). doi:10.1145/290179.290180

  2. 2.

    Arora, S., Raghavan, P., Rao, S.: Approximation Schemes for Euclidean K-medians and Related Problems. In: Proc.of the 30th Annual ACM Symposium on Theory of Computing, pp. 106–113. ACM. doi:10.1145/276698.276718(1998)

  3. 3.

    Bateni, M., Hajiaghayi, M.: Euclidean Prize-Collecting Steiner Forest. Algorithmica 62(3-4), 906–929 (2012). doi:10.1007/s00453-011-9491-8

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Borradaile, G., Klein, P.N., Mathieu, C.: A Polynomial-Time Approximation Scheme for Euclidean Steiner Forest. ACM Trans. Algorithm. 11(3), 19:1–19:20 (2015). doi:10.1145/2629654

    MathSciNet  Google Scholar 

  5. 5.

    Das, A., Mathieu, C.: A Quasi-Polynomial Time Approximation Scheme for Euclidean Capacitated Vehicle Routing. In: Proc.Of the 21St Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 390–403 (2010)

  6. 6.

    Du, D.Z., Hwang, F.: A proof of the Gilbert-Pollak conjecture on the Steiner ratio. Algorithmica 7(1-6), 121–135 (1992). doi:10.1007/BF01758755

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman (1979)

  8. 8.

    Gilbert, E.N., Pollak, H.O.: Steiner minimal trees. SIAM J. Appl. Math. 16(1), 1–29 (1968). doi:10.1137/0116001

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Goemans, M.X., Williamson, D.P.: A General Approximation Technique for Constrained Forest Problems. SIAM J. Comput. 24(2), 296–317 (1995). doi:10.1137/S0097539793242618

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Gupta, A., Kleinberg, J., Kumar, A., Rastogi, R., Yener, B.: Provisioning a Virtual Private Network: A Network Design Problem for Multicommodity Flow. In: Proc.of the 33th Annual ACM Symposium on Theory of Computing, pp. 389–398. doi:10.1145/380752.380830 (2001)

  11. 11.

    Karloff, H.J.: How Long Can a Euclidean Traveling Salesman Tour Be? SIAM J. Discret. Math. 2(1), 91–99 (1989). doi:10.1137/0402010

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Kolliopoulos, S.G., Rao, S.: A Nearly Linear-Time Approximation Scheme for the Euclidean k-Median Problem. SIAM J. Comput. 37(3), 757–782 (2007). doi:10.1137/S0097539702404055

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Meira, L.A.A., Miyazawa, F.K.: A Continuous Facility Location Problem and Its Application to a Clustering Problem. In: Proc.Of the 2008 ACM Symposium on Applied Computing. pp. 1826–1831 (2008)

  14. 14.

    Mitchell, J.S.B.: Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems. SIAM J. Comput. 28(4), 1298–1309 (1999). doi:10.1137/S0097539796309764

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Remy, J., Steger, A.: Approximation Schemes for Node-Weighted Geometric Steiner Tree Problems. Algorithmica 55(1), 240–267 (2007). doi:10.1007/s00453-007-9114-6

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Shmoys, D.B., Tardos, E., Aardal, K.: Approximation Algorithms for Facility Location Problems (Extended Abstract). In: Proc.of the 39th Annual ACM Symposium on Theory of Computing. pp. 265–274. doi:10.1145/258533.258600(1997)

  17. 17.

    Swamy, C., Kumar, A.: Primal-Dual Algorithms for Connected Facility Location Problems. Algorithmica 40(4), 245–269 (2004). doi:10.1007/s00453-004-1112-3

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Rafael C. S. Schouery.

Additional information

This work was partially supported by grants 2013/21744-8 , 2014/14209-1 and 2013/21744-8, São Paulo Research Foundation (FAPESP) and grants 311499/2014-7 and 477692/2012-5, National Council for Scientific and Technological Development (CNPq).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Miyazawa, F.K., C. Pedrosa, L.L., S. Schouery, R.C. et al. A PTAS for the Geometric Connected Facility Location Problem. Theory Comput Syst 61, 871–892 (2017). https://doi.org/10.1007/s00224-017-9749-x

Download citation


  • Connected facility location problem
  • Geometric problem
  • Polynomial-time approximation scheme
  • Prize-collecting