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On the sample monotonization problem

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Abstract

The problem of finding a maximal subsample in a training sample consisting of the pairs “object-answer” that does not violate monotonicity constraints is considered. It is proved that this problem is NP-hard and that it is equivalent to the problem of finding a maximum independent set in special directed graphs. Practically important cases in which a partial order specified on the set of answers is a complete order or has dimension two are considered in detail. It is shown that the second case is reduced to the maximization of a quadratic convex function on a convex set. For this case, an approximate polynomial algorithm based on linear programming theory is proposed.

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References

  1. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness (Freeman, San Francisco, 1979; Mir, Moscow, 1982).

    MATH  Google Scholar 

  2. A. Schrijver, Theory of Linear and Integer Programming (Wiley, Chichester, 1986; Mir, Moscow, 1991).

    MATH  Google Scholar 

  3. L. G. Khachian, “A Polynomial Algorithm in Linear Programming,” Dokl. Akad. Nauk SSSR 244, 1093–1096 (1979) [Soviet Math. Dokl. 20, 191–194 (1979)].

    MathSciNet  Google Scholar 

  4. S. A. Cook, “The Complexity of Theorem Proving Procedures,” in Proc. of the 3rd ACM Symposium on the Theory of Computing, 1971.

  5. J. Edmonds and R. M. Karp, “Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems,” J. ACM 19(2), 248–264 (1972).

    Article  MATH  Google Scholar 

  6. M. Grotshel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization (Springer, Berlin, 1988).

    Google Scholar 

  7. D. S. Hochbaum, “Approximation Algorithms for the Set Covering and Vertex Cover Problems,” SIAM J. Comput. 11, 555–556 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  8. R. H. Mohring, “Algorithmic Aspects of Comparability Graphs and Interval Graphs,” in Graphs and Order, Ed. by I. Rival (Reidel, Boston, 1985), pp. 41–101.

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Authors and Affiliations

  1. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

    R. S. Takhanov

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  1. R. S. Takhanov
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Correspondence to R. S. Takhanov.

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Original Russian Text © R.S. Takhanov, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 7, pp. 1327–1333.

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Takhanov, R.S. On the sample monotonization problem. Comput. Math. and Math. Phys. 50, 1260–1266 (2010). https://doi.org/10.1134/S0965542510070146

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  • DOI: https://doi.org/10.1134/S0965542510070146

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