Abstract
The NP-completeness is proved of the problem of choosing some subset of “similar” vectors. One of the variants of the a posteriori (off-line) noise-proof detection problem of an unknown repeating vector in a numeric sequence can be reduced to this problem in the case of additive noise. An approximation polynomial algorithm with a guaranteed performance bound is suggested for this problem in the case of a fixed space dimension.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. E. Baburin, E. Kh. Gimadi, N. I. Glebov, and A. V. Pyatkin, “The Problem of Finding a Subset of Vectors with the Maximum TotalWeight,” Diskret. Anal. Issled. Oper. Ser. 2, 14(1), 32–42 (2007) [J. Appl. Indust. Math. 2 (1), 32–38 (2008)].
E. Kh. Gimadi, A. V. Kel’manov, M. A. Kel’manova, S. A. Khamidullin, “A Posteriori Detection of a Quasi-Periodic Fragment in a Number Sequence Under a Given Number of Repeats,” Sibirsk. Zh. Indust. Mat. 9(1), 55–74 (2006).
A. V. Kel’manov, “On Some Polynomially Solvable and NP-Hard Problems of Analysis and Recognition of Sequences with Quasi-Periodic Structure,” in Proceedings of the 13th Russian Conference “Mathematical Methods of Image Recognition” (Zelenogorsk, September 30–October 6, 2007) (MAKS Press, Moscow, 2007), pp. 261–264.
A. V. Kel’manov, “Polynomially Solvable and NP-Hard Variants of the Problem of the Optimal Finding of a Repeating Fragment in a Number Sequence,” in Proceedings of the Russian Conference “Discrete Optimization and Operation Research” (Vladivostok, September 7–14, 2007) (Inst.Mat., Novosibirsk, 2007), pp. 46–50 (http://math.nsc.ru/conference/door07/DOOR_abstracts.pdf).
A.V. Kel’manov and S. A. Khamidullin, “A Posteriori Detection of a Given Number of Identical Subsequences in a Quasi-Periodic Sequence,” Zh. Vychisl. Mat. i Mat. Fiz. 41(5), 807–820 (2001) [Comput.Math.Math. Phys. 41 (5), 762–774 (2001)].
D. Aloise and P. Hansen, “On the Complexity of Minimum Sum-of-Squares Clustering,” Les Cahiers du GERAD, G-2007-50 (2007).
P. Drineas, A. Frieze, R. Kannan, S. Vempala, and V. Vinay, “Clustering Large Graphs via the Singular Value Decomposition,” Machine Learning 56, 9–33 (2004).
M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, CA, 1979).
A. V. Kel’manov and B. Jeon, “A Posteriori Joint Detection and Discrimination of Pulses in a Quasi-Periodic Pulse Train,” IEEE Trans. Signal Process. 52(3), 1–12 (2004).
A. V. Kel’manov, S. A. Khamidullin, and L. V. Okol’nishnikova, “A Posteriori Detection of Identical Subsequences in a Quasi-Periodic Sequence,” Pattern Recognition and Image Analysis 12(4), 438–447 (2002).
J. MacQueen, “Some Methods for Classification and Analysis of Multivariate Observations,” in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (1967), pp. 281–296.
A. Wald, Sequential Analysis (Wiley, New York, 1947).
The QPSLab (Quasi-Periodic Sequence Laboratory) System, http://math.nsc.ru/~serge/qpsl/.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Kel’manov, A.V. Pyatkin, 2008, published in Diskretnyi Analiz i Issledovanie Operatsii, 2008, Vol. 15, No. 5, pp. 20–34.
Rights and permissions
About this article
Cite this article
Kel’manov, A.V., Pyatkin, A.V. On a version of the problem of choosing a vector subset. J. Appl. Ind. Math. 3, 447–455 (2009). https://doi.org/10.1134/S1990478909040036
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478909040036