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p-Center Problems

  • Hatice ÇalıkEmail author
  • Martine Labbé
  • Hande Yaman
Chapter
  • 60 Downloads

Abstract

A p-center is a minimax solution that consists of a set of p points minimizing the maximum distance between a demand point and a closest point belonging to that set. We present different variants of this problem. We review special polynomial cases, determine the complexity of the problems and present mixed integer linear programming formulations, exact algorithms and heuristics. Several extensions are also reviewed.

Keywords

p-Center Minimax facility location Location in public sector 

Notes

Acknowledgements

The research of the second author is supported by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office and the research of the third author is supported by the Turkish Academy of Sciences.

References

  1. Albareda-Sambola M, Díaz JA, Fernández E (2010) Lagrangean duals and exact solution to the capacitated p-center problem. Eur J Oper Res 201:71–81CrossRefMathSciNetzbMATHGoogle Scholar
  2. Averbakh I (1997) On the complexity of a class of robust location problems. Working Paper, Western Washington University, BellinghamGoogle Scholar
  3. Averbakh I, Berman O (1997) Minimax regret p-center location on a network with demand uncertainty. Locat. Sci. 5:247–254CrossRefzbMATHGoogle Scholar
  4. Averbakh I, Berman O (2000) Algorithms for the robust 1-center problem on a tree. Eur J Oper Res 123:292–302CrossRefMathSciNetzbMATHGoogle Scholar
  5. Bar-Ilan J, Kortsarz G, Peleg D (1993) How to allocate network centers. J Algorithm 15:385–415CrossRefMathSciNetzbMATHGoogle Scholar
  6. Beasley JE (1990) OR-library: distributing test problems by electronic mail. J Oper Res Soc 41:1069–1072CrossRefGoogle Scholar
  7. Berge B (1967) Théorie des graphes et ses applications, Dunod, PariszbMATHGoogle Scholar
  8. Berman O, Drezner Z (2008) A new formulation for the conditional p-median and p-center problems. Oper Res Lett 36:481–483CrossRefMathSciNetzbMATHGoogle Scholar
  9. Berman O, Simchi-Levi D (1990) Conditional location problems on networks. Transp Sci 24:77–78CrossRefMathSciNetzbMATHGoogle Scholar
  10. Bozkaya B, Tansel B (1998) A spanning tree approach to the absolute p-center problem. Locat. Sci. 6:83–107CrossRefGoogle Scholar
  11. Çalık H (2013) Exact solution methodologies for the p-center problem under single and multiple allocation strategies. Ph.D. Thesis, Bilkent University, AnkaraGoogle Scholar
  12. Calik H, Tansel BC (2013) Double bound method for solving the p-center location problem. Comput Oper Res 40:2991–2999CrossRefMathSciNetzbMATHGoogle Scholar
  13. Chechik S, Peleg D (2012) The fault tolerant capacitated k-center problem. Theor Comput Sci 566:12–25CrossRefMathSciNetzbMATHGoogle Scholar
  14. Chen D, Chen R (2013) Optimal algorithms for the α-neighbor p-center problem. Eur J Oper Res 225:36–43CrossRefMathSciNetzbMATHGoogle Scholar
  15. Daskin MS (2013) Network and discrete location: models, algorithms, and applications, 2nd edn. Wiley, HobokenzbMATHGoogle Scholar
  16. Drezner Z (1989) Conditional p-center problems. Transp. Sci 23:51–53CrossRefMathSciNetzbMATHGoogle Scholar
  17. Dyer ME, Frieze AM (1985) A simple heuristic for the p-center problem. Oper Res Lett 3:285–288CrossRefMathSciNetzbMATHGoogle Scholar
  18. Elloumi S, Labbé M, Pochet Y (2004) A new formulation and resolution method for the p-center problem. INFORMS J Comput 16:84–94CrossRefMathSciNetzbMATHGoogle Scholar
  19. Espejo I, Marín A, Rodríguez-Chía AM (2015) Capacitated p-center problem with failure foresight. Eur J Oper Res 247:229–244CrossRefMathSciNetzbMATHGoogle Scholar
  20. Fernandes CG, de Paula SP, Pedrosa LL (2018) Improved approximation algorithms for capacitated fault-tolerant k-center Algorithmica 80:1041–1072Google Scholar
  21. Garcia-Diaz J, Sanchez-Hernandez J, Menchaca-Mendez R, Menchaca-Mendez R (2017) When a worse approximation factor gives better performance: a 3-approximation algorithm for the vertex k-center problem. J Heuristics 23:349–366CrossRefGoogle Scholar
  22. Garfinkel R, Neebe A, Rao M (1977) The m-center problem: minimax facility location. Manag Sci 23:1133–1142CrossRefzbMATHGoogle Scholar
  23. Goldman AJ (1972) Minimax location of a facility in a network. Transp Sci 6:407–418CrossRefMathSciNetGoogle Scholar
  24. Hakimi SL (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Oper Res 12:450–459CrossRefzbMATHGoogle Scholar
  25. Hakimi SL (1965) Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper Res 13:462–475CrossRefMathSciNetzbMATHGoogle Scholar
  26. Handler GY (1973) Minimax location of a facility in an undirected tree network. Transp Sci 7:287–293CrossRefGoogle Scholar
  27. Hansen P, Labbé M, Nicloas B (1991) The continuous center set of a network. Discrete Appl Math 30:181–195CrossRefMathSciNetzbMATHGoogle Scholar
  28. Hochbaum DS, Shmoys DB (1985) A best possible heuristic for the k-center problem. Math Oper Res 10:180–184CrossRefMathSciNetzbMATHGoogle Scholar
  29. Hsu W-L, Nemhauser GL (1979) Easy and hard bottleneck location problems. Discrete Appl Math 1:209–215CrossRefMathSciNetzbMATHGoogle Scholar
  30. Ilhan T, Pınar MÇ (2001) An efficient exact algorithm for the vertex p-center problem. Bilkent University, Department of Industrial Engineering, Technical Report. http://www.ie.bilkent.edu.tr/~mustafap/pubs
  31. Irawan CA, Salhi S, Drezner Z (2016) Hybrid meta-heuristics with VNS and exact methods: application to large unconditional and conditional vertex p-centre problems. J Heuristics 22:507–537CrossRefGoogle Scholar
  32. Jaeger M, Goldberg J (1994) A polynomial algorithm for the equal capacity p-center problem on trees. Transp. Sci 28:167–175CrossRefMathSciNetzbMATHGoogle Scholar
  33. Kariv O, Hakimi SL (1979) An algorithmic approach to network location problems. I: the p-centers. SIAM J Appl Math 37:513–538CrossRefMathSciNetzbMATHGoogle Scholar
  34. Khuller S, Sussmann YJ (2000) The capacitated k-center problem. SIAM J Discrete Math 13:403–418CrossRefMathSciNetzbMATHGoogle Scholar
  35. Khuller S, Pless R, Sussmann YJ (2000) Fault tolerant K-center problems. Theor Comput Sci 242:237–245CrossRefMathSciNetzbMATHGoogle Scholar
  36. Krumke OS (1995) On a generalization of the p-center problem. Inf Process Lett 56:67–71CrossRefMathSciNetGoogle Scholar
  37. Lorena LAN, Senne ELF (2004) A column generation approach to capacitated p-median problems. Comput Oper Res 31:863–876CrossRefMathSciNetzbMATHGoogle Scholar
  38. Martínez-Merino LI, Albareda-Sambola, M, Rodríguez-Chía AM (2017) The probabilistic p-center problem: planning service for potential customers. Eur J Oper Res 262:509–520CrossRefMathSciNetzbMATHGoogle Scholar
  39. Martinich JS (1988) A vertex-closing approach to the p-center problem. Nav Res Log 35:185–201CrossRefMathSciNetzbMATHGoogle Scholar
  40. Megiddo N (1983) Linear-time algorithms for linear programming in R3 and related problems. SIAM J Comput 12:759–776CrossRefMathSciNetzbMATHGoogle Scholar
  41. Mihelič J, Robič B (2005) Solving the k-center problem efficiently with a dominating set algorithm. J Comput Inf Technol 13:225–233CrossRefGoogle Scholar
  42. Minieka E (1970) The m-center problem. SIAM Rev 12:138–139CrossRefMathSciNetzbMATHGoogle Scholar
  43. Minieka E (1980) Conditional centers and medians on a graph. Networks 10:265–272CrossRefMathSciNetGoogle Scholar
  44. Mladenović N, Labbé M, Hansen P (2003) Solving the p-center problem with tabu search and variable neighborhood search. Networks 42:48–64CrossRefMathSciNetzbMATHGoogle Scholar
  45. Özsoy, FA, Pınar, MÇ (2006) An exact algorithm for the capacitated vertex p-center problem. Comput Oper Res 33:1420–1436CrossRefMathSciNetzbMATHGoogle Scholar
  46. Pullan W (2008) A memetic genetic algorithm for the vertex p-center problem. Evol Comput 16:417–436CrossRefGoogle Scholar
  47. Quevedo-Orozco DR, Ríos-Mercado RZ (2015) Improving the quality of heuristic solutions for the capacitated vertex p-center problem through iterated greedy local search with variable neighborhood descent. Comput Oper Res 62:133–144CrossRefMathSciNetzbMATHGoogle Scholar
  48. Reinelt G (1991) TSPLIB - a traveling salesman problem library. ORSA J Comput 3:376–384CrossRefzbMATHGoogle Scholar
  49. Salhi S, Al-Khedhairi A (2010) Integrating heuristic information into exact methods: the case of the vertex p-centre problem. J Oper Res Soc 61:1619–1631CrossRefzbMATHGoogle Scholar
  50. Scapparra MP, Pallotino S, Scutella MG (2004) Large-scale local search heuristics for the capacitated vertex p-center problem. Networks 43:241–255CrossRefMathSciNetGoogle Scholar
  51. Tamir A (1987) On the solution value of the continuous p-center location problem on a graph. Math Oper Res 12:340–349CrossRefMathSciNetzbMATHGoogle Scholar
  52. Tamir A (1988) Improved complexity bounds for center location problems on networks by using dynamic data structures. SIAM J Discrete Math 1:377–396CrossRefMathSciNetzbMATHGoogle Scholar
  53. Tansel BÇ (2011) Discrete center problems. In: Eiselt HA, Marianov V (eds) Foundations of location analysis. Springer, New York, pp 79–106CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer Science, CODeSKU LeuvenGentBelgium
  2. 2.Department of Computer ScienceUniversité Libre de BruxellesBrusselsBelgium
  3. 3.Inria Lille-Nord EuropeLille, Villeneuve d’AscqFrance
  4. 4.Faculty of Economics and BusinessORSTAT, KU LeuvenLeuvenBelgium
  5. 5.Department of Industrial EngineeringBilkent UniversityAnkaraTurkey

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