Discrete & Computational Geometry

, Volume 14, Issue 4, pp 463–479 | Cite as

Almost optimal set covers in finite VC-dimension

  • H. Brönnimann
  • M. T. Goodrich


We give a deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous Θ(log⋎X⋎) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.


Computational Geometry Approximation Factor Weighted Case Greedy Method Boolean Matrix 
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  1. 1.
    E. M. Arkin, H. Meijer, J. S. B. Mitchell, D. Rappaport, and S. S. Skiena. Decision trees for geometric models.Proc. 9th Ann. ACM Symp. on Computational Geometry, pp. 369–378, 1993.Google Scholar
  2. 2.
    P. Assouad. Densité et dimension.Ann. Institut Fourier (Grenoble), 3:232–282, 1983.MathSciNetGoogle Scholar
  3. 3.
    E. Baum, On learning the union of halfspaces.J. Complexity, 6:67–101, 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Bellare, S. Goldwasser, C. Lund, and A. Russel. Efficient probabilistically checkable proofs and applications to approximation.Proc. 25th Ann. ACM Symp. on Theory of Computing, pp. 294–304, 1993.Google Scholar
  5. 5.
    B. Berger, J. Rompel, and P. W. Shor. Efficient NC algorithms for set cover with applications to learning and geometry.Proc. 30th Ann. IEEE Symp. on Foundations of Computer Science, pp. 54–59, 1989.Google Scholar
  6. 6.
    A. Blum and R. Rivest. Training a 3-node neural network is NP-complete.Proc. 1st Workshop on Computer Learning Theory, pp. 9–18, 1988.Google Scholar
  7. 7.
    A. Blumer, A. Ehrenfeucht, D. Haussler, and M. Warmuth. Classifying learnable geometric concepts with the Vapnik-Chervonenkis dimension.J. Assoc. Comput. Mach., 36:929–965, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    H. Brönnimann, Almost optimal polyhedral separators.Proc. 10th Ann. ACM Symp. on Computational Geometry, pp. 393–394, 1994. Accompanying video.Google Scholar
  9. 9.
    H. Brönnimann. An implementation ofmin hitting set heuristics. Manuscript, 1994.Google Scholar
  10. 10.
    H. Brönnimann, B. Chazelle, and J. Matoušek. Product range spaces, sensitive sampling, and derandomization.Proc. 34th Ann. IEEE Symp. on Foundations of Computer Science (FOCS 93), pp. 400–409, 1993Google Scholar
  11. 11.
    B. Chazelle. An optimal convex hull algorithm and new results on cuttings.Discrete Comput. Geom., 10:377–409, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    B. Chazelle. Cutting hyperplanes for divide-and-conquer.Discrete Comput. Geom., 9(2):145–158, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, and M. Sharir. Improved bounds on weak ∈-nets for convex sets.Proc. 25th Ann. ACM Symp. on Theory of Computing (STOC 93), pp. 495–504, 1993.Google Scholar
  14. 14.
    B. Chazelle and J. Friedman. A deterministic view of random sampling and its use in geometry.Combinatorica, 10(3):229–249, 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    B. Chazelle and J. Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension.Proc. 4th ACM-SLAM Symp. on Discrete Algorithms, pp. 281–290, 1993.Google Scholar
  16. 16.
    B. Chazelle and E. Welzl. Quasi-optimal range searching in spaces of finite VC-dimension.Discrete Comput. Geom., 4:467–489, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    V. Chvátal. A greedy heuristic for the set-covering problem.Math. Oper. Res., 4:233–235, 1979.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    K. L. Clarkson, A Las Vegas algorithm for linear programming when the dimension is small.Proc. 29th Ann. IEEE Symp. on Foundations of Computer Science, pp. 452–456, 1988.Google Scholar
  19. 19.
    K. L. Clarkson. Algorithms for polytope covering and approximation.Proc. 3rd Workshop on Algorithms and Data Structures, pp. 246–252.Lecture Notes in Computer Science, vol. 709. Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
  20. 20.
    K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II.Discrete Comput. Geom., 4:387–421, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    T. H. Cormen, C. E. Leiserson, and R. L. Rivest.Introduction to Algorithms. MIT Press, Cambridge, MA, 1990.zbMATHGoogle Scholar
  22. 22.
    G. Das. Approximation schemes in computational geometry. Ph.D. thesis, University of Wisconsin, 1990.Google Scholar
  23. 23.
    M. R. Garey and D. S. Johnson.Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York, 1979.zbMATHGoogle Scholar
  24. 24.
    M. T. Goodrich. Geometric partitioning made easier, even in parallel.Proc. 9th Ann. ACM Symp. on Computational Geometry. pp. 73–82, 1993.Google Scholar
  25. 25.
    D. Haussler and E. Welzl. Epsilon-nets and simplex range queries.Discrete Comput. Geom., 2:127–151, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D. Hochbaum. Approximation algorithms for the set covering and vertex cover problems.SIAM J. Comput., 11(3):555–556, 1982.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    D. S. Hochbaum and W. Maas. Approximation schemes for covering and packing problems in image processing and VLSI.J. Assoc. Comput. Mach., 32:130–136, 1985.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    D. S. Johnson. Approximation algorithms for combinatorial problems.J. Comput. System Sci., 9:256–278, 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    R. M. Karp, Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors,Complexity of Computer Computations, pp. 85–103. Plenum, New York, 1972.CrossRefGoogle Scholar
  30. 30.
    J. Komlós, J. Pach, and G. Woeginger. Almost tight bounds for ∈-nets.Discrete Comput. Geom., 7:163–173, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    N. Littlestone. Learning quickly when irrelevant attributes abound: a new linear-threshold algorithms.Proc. 28th IEEE Symp. on Foundations of Computer Science, pp. 68–77, 1987.Google Scholar
  32. 32.
    L. Lovász. On the ratio of optimal integral and fractional covers.Discrete Math., 13:383–390, 1975.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    C. Lund and M. Yannakakis. On the hardness of approximating minimization problems.Proc. 25th Ann. ACM Symp. on Theory of Computing, pp. 286–293, 1993.Google Scholar
  34. 34.
    J. Matoušek. Construction of ∈-nets.Discrete Comput. Geom., 5:427–448, 1990.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    J. Matoušek. Approximations and optimal geometric divide-and-conquer.Proc. 23rd Ann. ACM Symp. on Theory of Computing, pp. 505–511, 1991. Also to appear inJ. Comput. System Sci.Google Scholar
  36. 36.
    J. Matoušek. Cutting hyperplane arrangements.Discrete Comput. Geom., 6:385–406, 1991.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    J. Matoušek. Reporting points in halfspaces.Comput. Geom. Theory Appl., 2(3):169–186, 1992.CrossRefzbMATHGoogle Scholar
  38. 38.
    J. Matoušek. Efficient partition trees.Discrete Comput. Geom., 8:315–334, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    J. Matoušek. Range searching with efficient hierarchical cuttings.Discrete Comput. Geom., 10(2):157–182, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    J. Matoušek, R. Seidel, and E. Welzl. How to net a lot with little: small ∈-nets for disks and halfspaces.Proc. 6th Ann. ACM Symp. on Computational Geometry, pp. 16–22, 1990.Google Scholar
  41. 41.
    J. Matoušek, E. Welzl, and L. Wernisch. Discrepancy and ∈-approximations for bounded VC-dimension.Proc. 32nd Ann. IEEE Symp. on Foundations of Computer Science, pp. 424–430, 1991.Google Scholar
  42. 42.
    J. S. B. Mitchell and S. Suri. Separation and approximation of polyhedral surfaces.Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, pp. 296–306, 1992.Google Scholar
  43. 43.
    F. P. Preparata and M. I. Shamos,Computational Geometry: an Introduction. Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
  44. 44.
    S. Rajagopalan and V. Vazirani, Primal-dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs.Proc. 34th IEEE Ann. Symp. on Foundations of Computer Science, pp. 322–331, 1993.Google Scholar
  45. 45.
    N. Sauer. On the densities of families of sets.J. Combin. Theory, 13:145–147, 1972.CrossRefzbMATHGoogle Scholar
  46. 46.
    R. E. Tarjan.Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics. Philadelphia, PA, 1987.Google Scholar
  47. 47.
    V. N. Vapnik and A. Ya. Červonenkis. On the uniform convergence of relative frequencies of events to their probabilities.Theory Probab. Appl., 16:264–280, 1971.CrossRefzbMATHGoogle Scholar
  48. 48.
    E. Welzl. Partition trees for triangle counting and other range searching problems.Proc. 4th Ann. ACM Symp. on Computational Geometry, pp. 23–33, 1988.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • H. Brönnimann
    • 1
  • M. T. Goodrich
    • 2
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA
  2. 2.Department of Computer ScienceJohns Hopkins UniversityBaltimoreUSA

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