Approximation Schemes for Geometric NP-Hard Problems: A Survey

  • Sanjeev Arora
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2245)


Geometric optimization problems arise in many disciplines and are often NP- hard. One example is the famous Traveling Salesman Problem (TSP): given n points in the plane (more generally, in ℜd), find the shortest closed path that visits them all.


  1. 1.
    S. Arora. Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. JACM 45(5) 753–782, 1998. Preliminary versions in Proceedings of 37th IEEE Symp. on Foundations of Computer Science, 1996, and Proceedings of 38th IEEE Symp. on Foundations of Computer Science, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Arora and G. Karakostas. Approximation schemes for minimum latency problems. Proc. ACM Symposium on Theory of Computing, 1999.Google Scholar
  3. 3.
    S. Arora, P. Raghavan, and S. Rao. Approximation schemes for the Euclidean k-medians andrelatedproblems. In In Proc. 30th ACM Symposium on Theory of Computing, pp 106–113, 1998.Google Scholar
  4. 4.
    T. Asano, N. Katoh, H. Tamaki and T. Tokuyama. Covering Points in the Plane by k-Tours: Towards a Polynomial Time Approximation Scheme for General k. In Proc. 29th Annual ACM Symposium on Theory of Computing, pp 275–283, 1997.Google Scholar
  5. 5.
    A. Barvinok, S. Fekete, D. S. Johnson, A. Tamir, G. J. Woeginger, R. Woodroofe. The maximum traveling salesman problem. Merger of two papers in Proc. IPCO 1998 and Proc. ACM-SIAM SODA 1998.Google Scholar
  6. 6.
    M. Bern and D. Eppstein. Approximation algorithms for geometric problems. In [10].Google Scholar
  7. 7.
    N. Christofides. Worst-case analysis of a new heuristic for the traveling salesman problem. In J.F. Traub, editor, Symposium on new directions and recent results in algorithms and complexity, page 441. Academic Press, NY, 1976.Google Scholar
  8. 8.
    A. Czumaj and A. Lingas. A polynomial time approximation scheme for Euclidean minimum cost k-connectivity. Proc. 25th Annual International Colloquium on Automata, Languages and Programming, LNCS, Springer Verlag 1998.Google Scholar
  9. 9.
    W. Fernandez de la Vega and C. Kenyon. A randomized approximation scheme for metric MAX-CUT. Proc. 39th IEEE Symp. on Foundations of Computer Science, pp 468–471, 1998.Google Scholar
  10. 10.
    D. Hochbaum, ed. Approximation Algorithms for NP-hard problems. PWS Publishing, Boston, 1996.Google Scholar
  11. 11.
    S. G. Kolliopoulos and S. Rao. A nearly linear time approximation scheme for the Euclidean k-median problem. LNCS, vol.1643, pp 378–387, 1999.Google Scholar
  12. 12.
    E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys. The traveling salesman problem. John Wiley, 1985Google Scholar
  13. 13.
    J. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple PTAS for geometric k-MST, TSP, andrelatedproblems. SIAM J. Comp., 28, 1999. Preliminary manuscript, April 30, 1996. To appear in SIAM J. Computing.Google Scholar
  14. 14.
    S. Rao and W. Smith. Approximating geometric graphs via “spanners” and “banyans.” In Proc. 30th ACM Symposium on Theory of Computing, pp. 540–550, 1998.Google Scholar
  15. 15.
    A.Z. Zelikovsky. An 11/6-approximation algorithm for the network Steiner Problem. Algorithmica, 9:463–470, 1993.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Sanjeev Arora
    • 1
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA

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