Algorithmica

, Volume 57, Issue 2, pp 252–283

On the Planar Piecewise Quadratic 1-Center Problem

Article
  • 128 Downloads

Abstract

In this paper we introduce a minimax model unifying several classes of single facility planar center location problems. We assume that the transportation costs of the demand points to the serving facility are convex functions {Qi}, i=1,…,n, of the planar distance used. Moreover, these functions, when properly transformed, give rise to piecewise quadratic functions of the coordinates of the facility location. In the continuous case, using results on LP-type models by Clarkson (J. ACM 42:488–499, 1995), Matoušek et al. (Algorithmica 16:498–516, 1996), and the derandomization technique in Chazelle and Matoušek (J. Algorithms 21:579–597, 1996), we claim that the model is solvable deterministically in linear time. We also show that in the separable case, one can get a direct O(nlog n) deterministic algorithm, based on Dyer (Proceedings of the 8th ACM Symposium on Computational Geometry, 1992), to find an optimal solution. In the discrete case, where the location of the center (server) is restricted to some prespecified finite set, we introduce deterministic subquadratic algorithms based on the general parametric approach of Megiddo (J. ACM 30:852–865, 1983), and on properties of upper envelopes of collections of quadratic arcs. We apply our methods to solve and improve the complexity of a number of other location problems in the literature, and solve some new models in linear or subquadratic time complexity.

Keywords

Center location Quadratic programming LP-type models Parametric approach 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amenta, N.: Helly-type theorems and generalized linear programming. Ph.D. dissertation, University of California, Berkeley (1993) Google Scholar
  2. 2.
    Amenta, N.: Helly-type theorems and generalized linear programming. Discrete Comput. Geom. 12, 241–261 (1994) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Basu, S., Pollack, R., Roy, M.-F.: On the combinatorial and algebraic complexity of quantifier elimination. J. ACM 43, 1002–1045 (1996) MATHMathSciNetGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization (2001) Google Scholar
  5. 5.
    Berger, M.: Geometry, I and II. Serie Universitext. Springer, Berlin (2004). Corrected 3rd printing Google Scholar
  6. 6.
    Berman, O., Drezner, Z., Wesolowsky, G.O.: The collection depot location problem on networks. Nav. Res. Logist. 49, 292–318 (2002) MathSciNetGoogle Scholar
  7. 7.
    Blakey, J.: University Mathematics. Blakie and Son, London (1957) Google Scholar
  8. 8.
    Brimberg, J., Wesolowsky, G.O.: Locating facilities by minimax relative to closest points of demand areas. Comput. Oper. Res. 29, 625–636 (2002) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chan, A., Hearn, D.W.: A rectilinear distance round-trip location problem. Transp. Sci. 11, 107–123 (1977) CrossRefGoogle Scholar
  10. 10.
    Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. In: Proceedings of the 4th ACM Symposium on Discrete Algorithms, Austin, TX, pp. 281–290 (1993). Also J. Algorithms 21, 579–597 (1996) Google Scholar
  11. 11.
    Clarkson, K.L.: Linear programming in \(O(n3^{d^{2}})\) time. Inf. Process. Lett. 22, 21–24 (1986) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Clarkson, K.L.: Las Vegas algorithms for linear programming and integer programming when the dimension is small. J. ACM 42, 488–499 (1995) MATHMathSciNetGoogle Scholar
  13. 13.
    Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 34, 200–208 (1987) MathSciNetGoogle Scholar
  14. 14.
    Cole, R.: Parallel merge sort. SIAM J. Comput. 17, 770–785 (1988) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Drezner, Z., Wesolowsky, G.O.: On the collection depot location problem. Eur. J. Oper. Res. 130, 510–518 (2001) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Drezner, Z., Stein, G., Wesolowsky, G.O.: One-facility location with rectilinear tour distances. Nav. Res. Logist. Q. 32, 391–405 (1985) MATHCrossRefGoogle Scholar
  17. 17.
    Dyer, M.E.: On a multidimensional search procedure and its application to the Euclidean one-centre problem. SIAM J. Comput. 15, 725–738 (1986) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dyer, M.E.: A class of convex programs with applications to computational geometry. In: Proceedings of the 8th ACM Symposium on Computational Geometry, pp. 9–15 (1992) Google Scholar
  19. 19.
    Elzinga, J., Hearn, D.W.: The minimum covering sphere problem. Manag. Sci. 19, 96–104 (1972) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Elzinga, J., Hearn, D.W.: Geometrical solutions for some minimax location problems. Transp. Sci. 96, 379–394 (1972) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Fernández, F.R., Nickel, S., Puerto, J., Rodríguez-Chía, A.M.: Robustness in the Pareto-solutions for the multi-criteria minisum location problem. J. Multi-Criteria Dec. Anal. 10, 191–203 (2001) MATHCrossRefGoogle Scholar
  22. 22.
    Foul, A.: A 1-center problem on the plane with uniformly distributed demand points. Oper. Res. Lett. 34, 264–268 (2006) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Frenk, J.B., Gromicho, J., Zhang, S.: General models in min-max continuous location: Theory and solution techniques. J. Optim. Theory Appl. 89, 39–63 (1996) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Frenk, J.B., Gromicho, J., Zhang, S.: General models in min-max planar location: Checking optimality conditions. J. Optim. Theory Appl. 89, 65–87 (1996) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Goodrich, M.: Approximation algorithms to design parallel algorithms that may ignore processor allocation. In: Proceedings 32-nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 711–722 (1991) Google Scholar
  26. 26.
    Halman, N.: Discrete and lexicographic Helly theorems and their relations to LP-type problems. Ph.D. dissertation, Tel Aviv University (2004) Google Scholar
  27. 27.
    Halman, N.: On the algorithmic aspects of discrete and lexicographic Helly-type theorems and the discrete LP-type model. SIAM J. Comput. 38, 1–45 (2008) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lee, D.T., Wu, Y.F.: Geometric complexity of some location problems. Algorithmica 1, 193–212 (1986) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16, 498–516 (1996) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Megiddo, N.: Linear time algorithms for linear programming in R 3 and related problems. SIAM J. Comput. 12, 759–776 (1983) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30, 852–865 (1983) MATHMathSciNetGoogle Scholar
  32. 32.
    Megiddo, N.: Linear programming in linear time when dimension is fixed. J. ACM 31, 114–127 (1984) MATHMathSciNetGoogle Scholar
  33. 33.
    Megiddo, N.: On the ball spanned by balls. Discrete Comput. Geom. 4, 605–610 (1989) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Nickel, S., Puerto, J., Rodríguez-Chía, A.M.: An approach to location models involving sets as existing facilities. Math. Oper. Res. 28, 693–715 (2003) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Ogryczak, W., Tamir, A.: Mimimizing the sum of the k-largest functions in linear time. Inf. Process. Lett. 85, 117–122 (2003) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Ohsawa, Y.: A geometrical solution for quadratic bicriteria location models. Eur. J. Oper. Res. 114, 380–388 (1999) MATHCrossRefGoogle Scholar
  37. 37.
    Puerto, J., Tamir, A., Mesa, J.A., Perez-Brito, D.: Center location problems on tree graphs with subtree-shaped customers. Discrete Appl. Math. doi:10.1016/j.dam.2007.11022
  38. 38.
    Renegar, J.: A faster PSPACE algorithm for deciding the existential theory of the reals. In: Proc. 29th IEEE Symposium on Foundations of Computer Science, pp. 291–295 (1988) Google Scholar
  39. 39.
    Renegar, J.: On the computational complexity and geometry of the first order theory of the reals. J. Symb. Comput. 13, 255–352 (1992) MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  41. 41.
    Sharir, M., Agarwal, P.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1995) MATHGoogle Scholar
  42. 42.
    Tamir, A., Halman, N.: One-way and round-trip center location problems. Discrete Optim. 2, 168–184 (2005) MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Toledo, S.: Maximizing non-linear concave functions in fixed dimension. In: Proceedings 33rd Annual Symposium on Foundations of Computer Science, pp. 676–685 (1992) Google Scholar
  44. 44.
    Toledo, S.: Maximizing non-linear concave functions in fixed dimension. In: Pardalos, P.M. (ed.) Complexity in Numerical Computations, pp. 429–447. World Scientific, Singapore (1993) Google Scholar
  45. 45.
    Zemel, E.: An O(n) algorithm for the linear multiple choice knapsack problem and related problems. Inf. Process. Lett. 18, 123–128 (1984) MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Zhou, G., Toh, K.C., Sun, J.: Efficient algorithms for the smallest enclosing ball problem. Comput. Optim. Appl. 30, 147–160 (2005) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Facultad de MatemáticasUniversidad de SevillaSevillaSpain
  2. 2.Facultad de CienciasUniversidad de CádizCádizSpain
  3. 3.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations