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Using the Pseudo-Dimension to Analyze Approximation Algorithms for Integer Programming

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Book cover Algorithms and Data Structures (WADS 2001)

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Abstract

We prove approximation guarantees for randomized algorithms for packing and covering integer programs expressed in certain normal forms. The bounds are in terms of the pseudo-dimension of the matrix of the coefficients of the constraints and the value of the optimal solution; they are independent of the number of constraints and the number of variables. The algorithms take time polynomial in the length of the representation of the integer program and the value of the optimal solution. We establish a related result for a class we call the mixed covering integer programs, which contains the covering integer programs. We describe applications of these techniques and results to a generalization of Dominating Set motivated by distributed file sharing applications, to an optimization problem motivated by an analysis of boosting, and to a generalization of matching in hypergraphs.

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References

  1. M. Anthony and P. L. Bartlett. Neural Network Learning: Theoretical Foundations. Cambridge University Press, 1999.

    Google Scholar 

  2. N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler. Scale-sensitive dimensions, uniform convergence, and learnability. Journal of the Association for Computing Machinery, 44(4):616–631, 1997.

    MathSciNet  Google Scholar 

  3. B.S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the Association for Computing Machinery, 41:153–180, 1994.

    MATH  MathSciNet  Google Scholar 

  4. S. Ben-David, N. Cesa-Bianchi, D. Haussler, and P. M. Long. Characterizations of learnability for classes of 0,..., n-valued functions. Journal of Computer and System Sciences, 50(1):74–86, 1992.

    Article  MathSciNet  Google Scholar 

  5. A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. JACM, 36(4):929–965, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Bronnimann and M. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom., 14:263–279, 1995.

    Article  MathSciNet  Google Scholar 

  7. P. L. Bartlett and P. M. Long. Prediction, learning, uniform convergence, and scale-sensitive dimensions. Journal of Computer and System Sciences, 56(2):174–190, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. L. Bartlett, P. M. Long, and R. C. Williamson. Fat-shattering and the learnability of real-valued functions. Journal of Computer and System Sciences, 52(3):434–452, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Crescenzi and V. Kann. A compendium of NP optimization problems. See http://www.nada.kth.se/viggo/problemlist/compendium.html.

  10. R. M. Dudley. Central limit theorems for empirical measures. Annals of Probability, 6(6):899–929, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  11. M.R. Fellows. The Robertson-Seymour theorems: A survey of applications. Contemp. Math., 89:1–18, 1987.

    MathSciNet  Google Scholar 

  12. Y. Freund. Boosting a weak learning algorithm by majority. Information and Computation, 121(2):256–285, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  13. Y. Freund and R. E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 55(1):119–139, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. Haussler. Decision theoretic generalizations of the PAC model for neural net and other learning applications. Information and Computation, 100(1):78–150, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. Haussler. Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension. Journal of Combinatorial Theory, Series A, 69(2):217–232, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and G. Turán. Threshold circuits of bounded depth. Journal of Computer and System Sciences, 46:129–154, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. J. Kearns and R. E. Schapire. Efficient distribution-free learning of probabilistic concepts. Journal of Computer and System Sciences, 48(3):464–497, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  18. P.G. Kolaitis and M.N. Thakur. Logical definability of NP optimization problems. Information and Computation, 115(2):321–353, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  19. Y. Li, P. M. Long, and A. Srinivasan. Improved bounds on the sample complexity of learning. Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 309–318, 2000.

    Google Scholar 

  20. L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383–390, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.

    Google Scholar 

  22. B. K. Natarajan. On learning sets and functions. Machine Learning, 4:67–97, 1989.

    Google Scholar 

  23. M. Naor and R. M. Roth. Optimal file sharing in distributed networks. SIAM J. Comput., 24(1):158–183, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Pach and P.K. Agarwal. Combinatorial Geometry. John Wiley and Sons, 1995.

    Google Scholar 

  25. D. Pollard. Convergence of Stochastic Processes. Springer Verlag, 1984.

    Google Scholar 

  26. J.R. Quinlan. Some elements of machine learning. Proceedings of the Sixteenth International Conference on Machine Learning, pages 523–524, 1999.

    Google Scholar 

  27. P. Raghavan. Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Journal of Computer and System Sciences, 37:130–143, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  28. F.P. Ramsey. On a problem of formal logic. Proceedings of the London Mathematics Society, 30(2):264–286, 1930.

    Article  Google Scholar 

  29. S. Roy. Semantic complexity of relational queries and data independent data partitioning. Proceedings of the ACM SIGACT-SIGART-SIGMOD Annual Symposium on Principles of Database Systems, 1991.

    Google Scholar 

  30. P. Raghavan and C. Thompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica, 7(4):365–374, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  31. J. Shawe-Taylor, M. Anthony, and N. Biggs. Bounding the sample size with the Vapnik-Chervonenkis dimension. Discrete Applied Mathematics, 42:65–73, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  32. R. Schapire. The strength of weak learnability. Machine Learning, 5:197–227, 1990.

    Google Scholar 

  33. Robert E. Schapire, Yoav Freund, Peter Bartlett, and Wee Sun Lee. Boosting the margin: A new explanation for the effectiveness of voting methods. The Annals of Statistics, 26(5):1651–1686, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  34. A. Srinivasan. Improved approximation guarantees for packing and covering integer programs. SIAM Journal on Computing, 29:648–670, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Srinivasan. New approaches to covering and packing problems. Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, 2001.

    Google Scholar 

  36. M. Talagrand. Sharper bounds for Gaussian and empirical processes. Annals of Probability, 22:28–76, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  37. L. G. Valiant. A theory of the learnable. Communications of the ACM, 27(11):1134–1142, 1984.

    Article  MATH  Google Scholar 

  38. V. N. Vapnik. Estimation of Dependencies based on Empirical Data. Springer Verlag, 1982.

    Google Scholar 

  39. V. N. Vapnik. Inductive principles of the search for empirical dependences (methods based on weak convergence of probability measures). Proceedings of the 1989 Workshop on Computational Learning Theory, 1989.

    Google Scholar 

  40. V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264–280, 1971.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Genome Institute of Singapore, 1 Research Link IMA Building, National University of Singapore, Singapore, 117604, Republic of Singapore

    Philip M. Long

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  1. Philip M. Long
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Editors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Candada, K1S 5B6

    Frank Dehne  & Jörg-Rüdiger Sack  & 

  2. Center for Geometric Computing, Brown University, Providence, RI, 02912-1910, USA

    Roberto Tamassia

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© 2001 Springer-Verlag Berlin Heidelberg

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Long, P.M. (2001). Using the Pseudo-Dimension to Analyze Approximation Algorithms for Integer Programming. In: Dehne, F., Sack, JR., Tamassia, R. (eds) Algorithms and Data Structures. WADS 2001. Lecture Notes in Computer Science, vol 2125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44634-6_4

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  • DOI: https://doi.org/10.1007/3-540-44634-6_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42423-9

  • Online ISBN: 978-3-540-44634-7

  • eBook Packages: Springer Book Archive

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