Skip to main content
SpringerLink
Log in

Geometric knapsack problems

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We study a variety of geometric versions of the classical knapsack problem. In particular, we consider the following “fence enclosure” problem: given a setS ofn points in the plane with valuesv i ≥ 0, we wish to enclose a subset of the points with a fence (a simple closed curve) in order to maximize the “value” of the enclosure. The value of the enclosure is defined to be the sum of the values of the enclosed points minus the cost of the fence. We consider various versions of the problem, such as allowingS to consist of points and/or simple polygons. Other versions of the problems are obtained by restricting the total amount of fence available and also allowing the enclosure to consist of at mostM connected components. When there is an upper bound on the length of fence available, we show that the problem is NP-complete. We also provide polynomial-time algorithms for many versions of the fence problem when an unrestricted amount of fence is available.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilber, Geometric applications of a matrix searching algorithm,Algorithmica,2 (1987), 195–208.

    Google Scholar 

  2. M. Atallah, R. Cole, and M. T. Goodrich, Cascading divide-and-conquer,SIAM Journal on Computing,18 (1989), 499–532.

    Google Scholar 

  3. D. Bienstock and C. L. Monma, Optimal enclosing regions in planar graphs,Networks,19 (1989), 79–94.

    Google Scholar 

  4. J. E. Boyce, D. P. Dobkin, R. L. Drysdale, and L. J. Guibas, Finding extremal polygons,SIAM Journal on Computing,14 (1985), 134–147.

    Google Scholar 

  5. V. Capoyleas, G. Rote, and G. Woeginger, Geometric clusterings,Journal of Algorithms,12 (1991), 341–356.

    Google Scholar 

  6. J. S. Chang, Polygon Optimization Problems, Ph.D. Dissertation, Technical Report No. 240, Robotics Report No. 78, Department of Computer Science, New York University, August, 1986.

  7. J. S. Chang and C. K. Yap, A polynomial solution for the potato peeling problem,Discrete and Computational Geometry,1 (1986), 155–182.

    Google Scholar 

  8. B. Chazelle, The polygon containment problem,Advances in Computing Research, JAI Press, Greenwich, CT, 1983, pp. 1–33.

    Google Scholar 

  9. V. Chvatal,Linear Programming, Freeman, San Francisco, CA, 1983, pp. 374–380.

    Google Scholar 

  10. G. B. Dantzig,Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963.

    Google Scholar 

  11. P. Eades and D. Rappaport, The complexity of computing minimum separating polygons,Pattern Recognition Letters (1993), to appear.

  12. H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, Heidelberg, 1987.

    Google Scholar 

  13. H. Edelsbrunner, A. D. Robinson, and X. Shen, Covering Convex Sets with Non-Overlapping Polygons,Discrete Mathematics,81 (1990), 153–164.

    Google Scholar 

  14. D. Eppstein, M. Overmars, G. Rote, and G. Woeginger, Finding minimum area k-gons,Discrete and Computational Geometry,7 (1992), 45–58.

    Google Scholar 

  15. L. Fejes Toth, Some packing and covering theorems,Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum, A12 (1950), 62–67.

    Google Scholar 

  16. M. Fredman and R. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms,Journal of the ACM,34 (1987), 596–615.

    Google Scholar 

  17. M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, CA, 1978.

    Google Scholar 

  18. R. S. Garfinkel and G. L. Nemhauser,Integer Programming, Wiley, New York, 1972.

    Google Scholar 

  19. S. K. Ghosh and D. M. Mount, An output sensitive algorithm for computing visibility graphs,SIAM Journal on Computing,20 (1991), 888–910.

    Google Scholar 

  20. 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.

    Google Scholar 

  21. J. JáJá,An Introduction to Parallel Algorithms, Addison-Wesley, Reading, MA, 1991.

    Google Scholar 

  22. B. Korte and L. Lovasz, Polyhedral results on antimatroids,Proceedings of the New York Combinatorics Conference, to appear.

  23. D. T. Lee and Y. T. Ching, The power of geometric duality revisited,Information Processing Letters,21 (1985), 117–122.

    Google Scholar 

  24. S. Martello and P. Toth,Knapsack Problems-Algorithms and Computer Implementations, Wiley, New York, 1990.

    Google Scholar 

  25. D. Mount and R. Silverman, Packing and covering the plane with translates of a convex polygon,Journal of Algorithms,11 (1990), 564–580.

    Google Scholar 

  26. C. H. Papadimitrious and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ, 1982.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics and Statistics, State University of New York at Stony Brook, 11794-3600, Stony Brook, NY, USA

    Esther M. Arkin

  2. Institute for Advanced Computer Studies, University of Maryland, College Park, 20742, MD, USA

    Samir Khuller

  3. Applied Mathematics and Statistics, State University of New York at Stony Brook, 11794-3600, Stony Brook, NY, USA

    Joseph S. B. Mitchell

Authors
  1. Esther M. Arkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samir Khuller
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joseph S. B. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by D. T. Lee.

Esther M. Arkin was partially supported by NSF Grants DMC-8451984, ECSE-8857642, and DMS-8903304. Samir Khuller was partially supported by NSF Grant CCR-8906949. Part of this research was done while he was at Cornell University, and supported by NSF Grant DCR 85-52938, an IBM Graduate Fellowship, and PYI matching funds from AT&T Bell Labs and Sun Microsystems. Joseph S. B. Mitchell was partially supported by NSF Grants IRI-8710858, ECSE-8857642, a grant from Hughes Research Labs, and by Air Force Office of Scientific Research Contract AFOSR-91-0328.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arkin, E.M., Khuller, S. & Mitchell, J.S.B. Geometric knapsack problems. Algorithmica 10, 399–427 (1993). https://doi.org/10.1007/BF01769706

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01769706

Key words

Search

Navigation