Approximation of geometric dispersion problems

Extended abstract
  • Christoph Baur
  • Sándor P. Fekete
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1444)


We consider problems of distributing a number of points within a connected polygonal domain P, such that the points are “far away” from each other. Problems of this type have been considered before for the case where the possible locations form a discrete set. Dispersion problems are closely related to packing problems. While Hochbaum and Maass (1985) have given a polynomial time approximation scheme for packing, we show that geometric dispersion problems cannot be approximated arbitrarily well in polynomial time, unless P=NP. We give a 2/3 approximation algorithm for one version of the geometric dispersion problem. This algorithm is strongly polynomial in the size of the input, i.e., its running time does not depend on the area of P. We also discuss extensions and open problems.


Approximation Scheme Knapsack Problem Binary Search Approximation Factor Simple Polygon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    B. S. Baker, D. J. Brown and H. K. Katseff. A 5/4 algorithm for two-dimensional packing, Journal of Algorithms, 2, 1981, 348–368.zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Christoph Baur. Packungs-und Dispersionsprobleme. Diplomarbeit, Mathematisches Institut, UniversitÄt zu Köln, 1997.Google Scholar
  3. 3.
    J. E. Beasley. An exact two-dimensional non-guillotine cutting tree search procedure, Operations Research, 33, 1985, 49–64.zbMATHMathSciNetGoogle Scholar
  4. 4.
    M. Bern and D. Eppstein. Clustering. Section 8.5 of the chapter Approximation algorithms for geometric problems, in: D. Hochbaum (ed.): Approximation Algorithms for NP-hard Problems. PWS Publishing, 1996.Google Scholar
  5. 5.
    B. K. Bhattacharya and M. E. Houle. Generalized maximum independent set for trees. To appear in: Journal of Graph Algorithms and Applications, 1997. Scholar
  6. 6.
    R. Chandrasekaran and A. Daughety. Location on tree networks: p-centre and n-dispersion problems. Mathematics of Operations Research, 6, 1981, 50–57.zbMATHMathSciNetGoogle Scholar
  7. 7.
    R. L. Church and R. S. Garfinkel. Locating an obnoxious facility on a network. Transportation Science, 12, 1978, 107–118.MathSciNetGoogle Scholar
  8. 8.
    P. Duchet, Y. Hamidoune, M. Las Vergnas, and H. Meyniel. Representing a planar graph by vertical lines joining different levels. Discrete Mathematics, 46, 1983, 319–321.zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    E. Erkut. The discrete p-dispersion problem. European Journal of Operational Research, 46, 1990, 48–60.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Erkut and S. Neumann. Comparison of four models for dispersing facilities. European Journal of Operations Research, 40, 1989, 275–291.zbMATHCrossRefGoogle Scholar
  11. 11.
    S. P. Fekete and J. Schepers. A new exact algorithm for general orthogonal d-dimensional knapsack problems. Algorithms—ESA '97, Springer Lecture Notes in Computer Science, vol. 1284, 1997, 144–156.CrossRefGoogle Scholar
  12. 12.
    S. P. Fekete and J. Schepers. On more-dimensional packing I: Modeling. Technical report, ZPR 97-288. Available at Scholar
  13. 13.
    S. P. Fekete and J. Schepers. On more-dimensional packing II: Bounds. Technical report, ZPR 97-289. Available at Scholar
  14. 14.
    S. P. Fekete and J. Schepers. On more-dimensional packing III: Exact Algorithms. Technical report, ZPR 97-290. Available at Scholar
  15. 15.
    R. J. Fowler, M. S. Paterson, and S. L. Tanimoto. Optimal packing and covering in the plane are NP-complete. Information Processing Letters, 12, 1981, 133–137.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Z. Füredi. The densest packing of equal circles into a parallel strip. Discrete & Computational Geometry, 1991, 95–106.Google Scholar
  17. 17.
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the theory of NP-Completeness. W. H. Freeman and Company, San Francisco, 1979.zbMATHGoogle Scholar
  18. 18.
    A. J. Goldmann and P. M. Dearing. Concepts of optimal locations for partially noxious facilities. Bulletin of the Operational Research Society of America. 23, B85, 1975.Google Scholar
  19. 19.
    R. L. Graham and B. D. Lubachevsky. Dense packings of equal disks in an equilateral triangle: from 22 to 34 and beyond. The Electronic Journal of Combinatorics, 2, 1995, #A1.Google Scholar
  20. 20.
    R. L. Graham and B. D. Lubachevsky. Repeated patterns of dense packings of equal disks in a square. The Electronic Journal of Combinatorics 3, 1996, #R16.Google Scholar
  21. 21.
    R. L. Graham, B. D. Lubachevsky, K. J. Nurmela, and P. R. J. östergård. Dense packings of congruent circles in a circle. Manuscript, 1996. Scholar
  22. 22.
    E. Hadjiconstantinou and N. Christofides. An exact algorithm for general, orthogonal, two-dimensional knapsack problems, European Journal of Operations Research, 83, 1995, 39–56.zbMATHCrossRefGoogle Scholar
  23. 23.
    D. S. Hochbaum and W. Maass. Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM, 32, 1985, 130–136.zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    C. Kenyon and E. Remila, Approximate strip packing. Proc. of the 37th Annual Symposium on Foundations of Computer Science (FOCS 96), 142–154.Google Scholar
  25. 25.
    D. Lichtenstein. Planar formulae and their uses. SIAM Journal on Computing, 11, 1982, 329–343.zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Z. Li and V. Milenkovic. A compaction algorithm for non-convex polygons and its application. Proc. of the Ninth Annual Symposium on Computational Geometry, 1993, 153–162.Google Scholar
  27. 27.
    B. D. Lubachevsky and R. L. Graham. Curved hexagonal packings of equal disks in a circle. Manuscript. Scholar
  28. 28.
    B. D. Lubachevsky, R. L. Graham, and F. H. Stillinger. Patterns and structures in disk packings. 3rd Geometry Festival, Budapest, Hungary, 1996. Manuscript. Scholar
  29. 29.
    C. D. Maranas, C. A. Floudas, and P. M. Pardalos. New results in the packing of equal circles in a square. Discrete Mathematics 142, 1995, 287–293.zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    M. V. Marathe, H. Breu, H. B. Hunt III, S. S. Ravi, and D. J. Rosenkrantz. Simple heuristics for unit disk graphs. Networks, 25, 1995, 59–68.zbMATHMathSciNetGoogle Scholar
  31. 31.
    S. Martello and P. Toth, Knapsack Problems — Algorithms and Computer Implementations, Wiley, Chichester, 1990.zbMATHGoogle Scholar
  32. 32.
    I.D. Moon and A. J. Goldman. Tree location problems with minimum separation. Transactions of the Institute of Industrial Engineers, 21, 1989, 230–240.Google Scholar
  33. 33.
    J. S. B. Mitchell, Y. Lin, E. Arkin, and S. Skiena. Stony Brook Computational Geometry Problem Seminar. Manuscript. Scholar
  34. 34.
    J. Neli\en. New approaches to the pallet loading problem. Technical Report, RWTH Aachen, Lehrstuhl für Angewandte Mathematik, 1993.Google Scholar
  35. 35.
    S. S. Ravi, D. J. Rosenkrantz, and G. K. Tayi. Heuristic and special case algorithms for dispersion problems. Operations Research, 42, 1994, 299–310.zbMATHCrossRefGoogle Scholar
  36. 36.
    D. J. Rosenkrantz, G. K. Tayi, and S. S. Ravi. Capacitated facility dispersion problems. Manuscript, submitted for publication, 1997. Scholar
  37. 37.
    P. Rosenstiehl and R. E. Tarjan. Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete & Computational Geometry, 1, 1986, 343–353.zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    J. Schepers. Exakte Algorithmen für orthogonale Packungsprobleme. Dissertation, Mathematisches Institut, UniversitÄt zu Köln, 1997.Google Scholar
  39. 39.
    A. Tamir. Obnoxious facility location on graphs. SIAM Journal on Discrete Mathematics, 4, 1991, 550–567.zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    M. Wottawa. Struktur und algorithmische Behandlung von praxisorientierten dreidimensionalen Packungsproblemen. Dissertation, Mathematisches Institut, UniversitÄt zu Köln, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Christoph Baur
    • 1
  • Sándor P. Fekete
    • 1
  1. 1.Center for Parallel ComputingUniversitÄt zu KölnKölnGermany

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