, Volume 9, Issue 4, pp 398–423 | Cite as

The searching over separators strategy to solve some NP-hard problems in subexponential time

  • R. Z. Hwang
  • R. C. Chang
  • R. C. T. Lee


In this paper we propose a new strategy for designing algorithms, called the searching over separators strategy. Suppose that we have a problem where the divide-and-conquer strategy can not be applied directly. Yet, also suppose that in an optimal solution to this problem, there exists a separator which divides the input points into two parts,Ad andCd, in such a way that after solving these two subproblems withAd andCd as inputs, respectively, we can merge the respective subsolutions into an optimal solution. Let us further assume that this problem is an optimization problem. In this case our searching over separators strategy will use a separator generator to generate all possible separators. For each separator, the problem is solved by the divide-and-conquer strategy. If the separator generator is guaranteed to generate the desired separator existing in an optimal solution, our searching over separators strategy will always produce an optimal solution. The performance of our approach will critically depend upon the performance of the separator generator. It will perform well if the total number of separators generated is relatively small. We apply this approach to solve the discrete EuclideanP-median problem (DEPM), the discrete EuclideanP-center problem (DEPC), the EuclideanP-center problem (EPC), and the Euclidean traveling salesperson problem (ETSP). We propose\(O(n^{o(\sqrt P )} )\) algorithms for the DEPM problem, the DEPC problem, and the EPC problem, and we propose an\(O(n^{o(\sqrt n )} )\) algorithm for the ETSP problem, wheren is the number of input points.

Key words

Computational geometry NP-hardness 


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  1. Aho, A. V., Hopcroft, J. E., and Ullman, J. D.,The Design and Analysis of Computer Algorithms, Bell Telephone Laboratories, Inc., New York, 1976.Google Scholar
  2. Bentley, J. L., Divide and Conquer Algorithms for Closest Point Problems in Multidimensional Space, Ph.D. Thesis, Department of Computer Science, University of North Carolina, 1976.Google Scholar
  3. Bentley, J. L., Multidimensional Divide-and-Conquer,Communications of the Association for Computing Machinery, Vol. 23, 1980, pp. 214–229.Google Scholar
  4. Delaunay, B., Sur la sphère vide,Izvestiya Akademii Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk, Vol. 7, 1934, pp. 793–800.Google Scholar
  5. Drezner, Z., TheP-Center Problem — Heuristics and Optimal Algorithms,Journal of the Operational Research Society, Vol. 35, No. 8, 1984, pp. 741–748.Google Scholar
  6. Drezner, Z., On the RectangularP-Center Problem,Naval Research Logistics, Vol. 34, 1987, pp. 229–234.Google Scholar
  7. Edelsbrunner, H.,Algorithms in Combinatorial Geometry, Springer-Verlag, New York, 1987.Google Scholar
  8. Held, M., and Karp, R. M., A Dynamic Programming Approach to Sequencing Problems,SIAM Journal on Applied Mathematics, Vol. 10, 1962, pp. 196–210.Google Scholar
  9. Horowitz, E., and Sahni, S.,Fundamentals of Computer Algorithms, Computer Science Press, Rockville, MD, 1978.Google Scholar
  10. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., and Shmoys, D. B.,The Traveling Salesman Problem — A Guided Tour of Combinatorial Optimization, Wiley-Interscience, New York, 1985.Google Scholar
  11. Lipton, R., and Tarjan, R. E., A Separator Theorem for Planar Graphs,SIAM Journal on Applied Mathematics, Vol. 36, No. 2, 1979, pp. 177–189.Google Scholar
  12. Megiddo, N., Linear-Time Algorithms for Linear Programming inR 3 and Related Problems,SIAM Journal on Computing, Vol. 12, No. 4, 1983, pp. 759–776.Google Scholar
  13. Megiddo, N., and Supowit, K. J., On the Complexity of Some Common Geometric Location Problems,SIAM Journal on Computing, Vol. 13, No. 1, 1984, pp. 182–196.Google Scholar
  14. Mehlhorn, K.,Data Structure and Algorithms 3: Multi-dimensional Search and Computational Geometry, Springer-Verlag, Berlin, 1984.Google Scholar
  15. Miller, G. L., Finding Small Simple Cycle Separators for 2-Connected Planar Graphs,Journal of Computer and System Sciences, Vol. 32, 1986, pp. 265–279.Google Scholar
  16. Nishizeki, T., and Chiba, N.,Planar Graphs, Theory and Algorithms, Elsevier, Amsterdam, 1988.Google Scholar
  17. Papadimitriou, C. H., Some Computational Problems Related to Database Concurrent Control,Proceedings of the Conference on Theoretical Computer Science, University of Waterloo, Waterloo, Ontario, 1977, pp. 275–282.Google Scholar
  18. Papadimitriou, C. H., Worst-Case and Probabilistic Analysis of a Geometric Location Problem,SIAM Journal on Computing, Vol. 10, 1981, pp. 542–557.Google Scholar
  19. Papadimitriou, C. H., and Steiglitz, K., Some Complexity Results for the Traveling Salesman Problem,Proceedings of the 8th Annual ACM Symposium on Theory of Computing, New York, 1976, pp. 1–9.Google Scholar
  20. Preparata, F. P., and Shamos M. I.,Computational Geometry, Springer-Verlag, New York, 1985.Google Scholar
  21. Smith, W. D., Studies in Computational Geometry Motivated by Mesh Generation, accepted byAlgorithmica, 1991.Google Scholar
  22. Voronoi, G., Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire: Sur quelques propriétés des formes quadratiques positives parfaites,Journal für die Reine und Angewandte Mathematik, Vol. 133, 1907, pp. 97–178.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • R. Z. Hwang
    • 1
  • R. C. Chang
    • 2
    • 3
  • R. C. T. Lee
    • 1
    • 3
  1. 1.Institute of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan, Republic of China
  2. 2.Institute of Computer ScienceNational Chiao Tung UniversityHsinchuTaiwan, Republic of China
  3. 3.Academia SinicaHsinchuTaiwan, Republic of China

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