# Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location

## Abstract

In this paper we introduce an optimization problem which involves maximization of the area of Voronoi regions of a set of points placed inside a circle. Such optimization goals arise in facility location problems consisting of both mobile and stationary facilities. Let ψ be a circular path through which mobile service stations are plying, and S be a set of n stationary facilities (points) inside ψ. A demand point p is served from a mobile facility plying along ψ if the distance of p from the boundary of ψ is less than that from any member in S. On the other hand, the demand point p is served from a stationary facility p i  ∈ S if the distance of p from p i is less than or equal to the distance of p from all other members in S and also from the boundary of ψ. The objective is to place the stationary facilities in S, inside ψ, such that the total area served by them is maximized. We consider a restricted version of this problem where the members in S are placed equidistantly from the center o of ψ. It is shown that the maximum area is obtained when the members in S lie on the vertices of a regular n-gon, with its circumcenter at o. The distance of the members in S from o and the optimum area increases with n, and at the limit approaches the radius and the area of the circle ψ, respectively. We also consider another variation of this problem where a set of n points is placed inside ψ, and the task is to locate a new point q inside ψ such that the area of the Voronoi region of q is maximized. We give an exact solution of this problem when n = 1 and a (1 − ε)-approximation algorithm for the general case.

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## References

1. Ahn, H.K., Cheng, S.W., Cheong, O., Golin, M., van Oostrum, R.: Competitive facility location along a highway. In: 7th Annual International Computing and Combinatorics Conference. LNCS, vol. 2108, pp. 237–246 (2001)

2. Alt, H., Cheong, O., Vigneron, A.: The Voronoi diagram of curved objects. Discrete Comput. Geom. 34, 439–453 (2005)

3. Apostol, T.: Mathematical Analysis. Narosa Publishing House (2002, reprinted)

4. Aurenhammer, F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. 23(3), 345–405 (1991)

5. Aurenhammer, F., Klein, R.: Voronoi Diagrams: Handbook of Computational Geometry. Elsevier, Amsterdam (2000)

6. Bhattacharya, B.B., Nandy, S.C.: New variations of the reverse facility location problem. In: Proc. 22nd Canadian Conference on Computational Geometry (2010, to appear)

7. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, New York (2000)

8. Cabello, S., Miguel Diáz-Báñez, J., Langerman, S., Seara, C., Ventura, I.: Facility location problems in the plane based on reverse nearest neighbor queries. Eur. J. Oper. Res. 202(1), 99–106 (2010)

9. Cheong, O., Efrat, A., Har-Peled, S.: On finding a guard that sees most and a shop that sells most. Discrete Comput. Geom. 37(4), 545–563 (2007)

10. Cheong, O., Har-Peled, S., Linial, N., Matoušek, J.: The one-round Voronoi game. Discrete Comput. Geom. Appl. 31(1), 125–138 (2004)

11. Dehne, F., Klein, R., Seidel, R.: Maximizing a Voronoi region: the convex case. Int. J. Comput. Geom. Appl. 15(5), 463–475 (2005)

12. Drezner, Z., Hamacher, H.W. (eds.): Facility Location: Applications and Theory. Springer, Heidelberg (2002)

13. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products. Academic, Orlando (1980)

14. Loney, S.L.: Elements of Coordinate Geometry. Macmillan and Co. (1953, reprinted)

15. Okabe, A., Boots, B., Sugihara, K.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York (1992)

16. Voronoi diagrams of points, segments, and arcs in 2D. http://www.cosy.sbg.ac.at/~held/projects/vroni/

17. Yap, C.K.: An O(n log n) time algorithm for the Voronoi diagram of a set of simple curve segments. Discrete Comput. Geom. 2, 365–393 (1987)

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Correspondence to Bhaswar B. Bhattacharya.

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Bhattacharya, B.B. Maximizing Voronoi Regions of a Set of Points Enclosed in a Circle with Applications to Facility Location. J Math Model Algor 9, 375–392 (2010). https://doi.org/10.1007/s10852-010-9142-0