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A fully polynomial-time approximation scheme for a sequence 2-cluster partitioning problem

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Abstract

We consider a strongly NP-hard problem of partitioning a finite sequence of points in Euclidean space into the two clustersminimizing the sum over both clusters of intra-cluster sums of squared distances from the clusters elements to their centers. The sizes of the clusters are fixed. The centroid of the first cluster is defined as the mean value of all vectors in the cluster, and the center of the second cluster is given in advance and equals 0. Additionally, the partition must satisfy the restriction that for all vectors in the first cluster the difference between the indices of two consequent points from this cluster is bounded from below and above by some given constants.We present a fully polynomial-time approximation scheme for the case of fixed space dimension.

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References

  1. A. E. Baburin, E. Kh. Gimadi, N. I. Glebov, and A. V. Pyatkin, “The Problem of Finding a Subset of Vectors with theMaximum TotalWeight,” Diskretn. Anal. Issled. Oper. Ser. 2, 14 (1), 32–42 (2007) [J. Appl. Indust. Math. 2 (1), 32–38 (2008. )].

    MathSciNet  MATH  Google Scholar 

  2. E. Kh. Gimadi, Yu. V. Glazkov, and I. A. Rykov, “On Two Problems of Choosing Some Subset of Vectors with Integer Coordinates That HasMaximum Norm of the Sum ofElements in EuclideanSpace,” Diskretn. Anal. Issled. Oper. 15 (4), 30–43 (2008) [J. Appl. Indust. Math. 3 (3), 343–352 (2009)].

    MATH  Google Scholar 

  3. E. Kh. Gimadi, A. V. Pyatkin, and I. A. Rykov, “On Polynomial Solvability of Some Problems of a Vector Subset Choice in an Euclidean Space of Fixed Dimension,” Diskretn. Anal. Issled. Oper. 15 (6), 11–19 (2008) [J. Appl. Indust. Math. 4 (1), 48–53 (2010)].

    MATH  Google Scholar 

  4. A. V. Dolgushev and A. V. Kel’manov, “An Approximation Algorithm for Solving a Problem of Cluster Analysis,” Diskretn. Anal. Issled. Oper. 18 (2), 29–40 (2011) [J. Appl. Indust. Math. 5 (4), 551–558 (2011)].

    MathSciNet  MATH  Google Scholar 

  5. A. V. Dolgushev, A. V. Kel’manov, and V. V. Shenmaier, “Polynomial-Time Approximation Scheme for a Problem of Partitioning a Finite Set into Two Clusters,” Trudy Inst. Mat. Mekh. Ural. Otdel. Ross. Akad. Nauk 21 (3), 100–109 (2015).

    MathSciNet  Google Scholar 

  6. A. V. Kel’manov, “Off-line Detection of aQuasi-Periodically Recurring Fragment in a Numerical Sequence,” Trudy Inst. Mat. Mekh. Ural. Otdel. Ross. Akad. Nauk 14 (2), 81–88 (2008) [Proc. Steklov Inst. Math. 263 (Suppl. 2), S84–S92 (2008)].

    MATH  Google Scholar 

  7. A. V. Kel’manov, “On the Complexity of SomeData Analysis Problems,” Zh. Vychisl. Mat. Mat. Fiz. 50 (11), 2045–2051 (2010) [Comput. Math. Math. Phys. 50 (11), 1941–1947 (2010)].

    MathSciNet  MATH  Google Scholar 

  8. A. V. Kel’manov, On the complexity of some cluster analysis problems, Zh. Vychisl. Mat. Mat. Fiz. 51 (11), 2106–2112 (2011) [Comput. Math. Math. Phys. 51 (11), 1983–1988 (2011)].

    MathSciNet  Google Scholar 

  9. A. V. Kel’manov and A. V. Pyatkin, “Complexity of Certain Problems of Searching for Subsets of Vectors and Cluster Analysis,” Zh. Vychisl. Mat. Mat. Fiz. 49 (11), 2059–2067 (2009) [Comput. Math. Math. Phys. 49 (11), 1966–1971 (2009)].

    MathSciNet  MATH  Google Scholar 

  10. A. V. Kel’manov and A. V. Pyatkin, “On Complexity of Some Problems of Cluster Analysis of Vector Sequences,” Diskretn. Anal. Issled. Oper. 20 (2), 47–57 (2013) [J. Appl. Indust. Math. 7 (3), 363–369 (2013)].

    MathSciNet  MATH  Google Scholar 

  11. A. V. Kel’manov and S. M. Romanchenko, “An FPTAS for a Vector Subset Search Problem,” Diskretn. Anal. Issled. Oper. 21 (3), 41–52 (2014) [J. Appl. Indust. Math. 8 (3), 329–336 (2014)].

    MathSciNet  MATH  Google Scholar 

  12. A. V. Kel’manov and S. A. Khamidullin, “Posterior Detection of a Given Number of Identical Subsequences in a Quasi-Periodic Sequence,” Zh. Vychisl. Mat. Mat. Fiz. 41 (5), 807–820 (2001) [Comput. Math. Math. Phys. 41 (5), 762–774 (2001)].

    MathSciNet  MATH  Google Scholar 

  13. A. V. Kel’manov and S. A. Khamidullin, “An Approximating Polynomial Algorithm for a Sequence Partitioning Problem,” Diskretn. Anal. Issled. Oper. 21 (1), 53–66 (2014) [J. Appl. Indust. Math. 8 (2), 236–244 (2014)].

    MATH  Google Scholar 

  14. A. V. Kel’manov and S. A. Khamidullin, “An Approximation Polynomial-Time Algorithm for a Sequence Bi-Clustering Problem,” Zh. Vychisl. Mat. Mat. Fiz. 55 (6), 1076–1085 (2015) [Comput. Math. Math. Phys. 55 (6), 1068–1076 (2015)].

    MathSciNet  MATH  Google Scholar 

  15. A. V. Kel’manov, S. A. Khamidullin, and V. I. Khandeev, “An Exact Pseudopolynomial Algorithm for a Sequence Bi-Clustering Problem,” in Abstracts of XV All-Russian Conference “Mathematical Programming and Applications,” Ekaterinburg, Russia, March 2–6, 2015 (Inst. Mat. Mekh. Ural. Otdel. Ross. Akad. Nauk, Ekaterinburg, 2015), pp. 139–140.

    Google Scholar 

  16. A. V. Kel’manov and V. I. Khandeev, “A Randomized Algorithm for Two-Cluster Partition of a Set of Vectors,” Zh. Vychisl. Mat. Mat. Fiz. 55 (2), 335–344 (2015) [Comput. Math. Math. Phys. 55 (2), 330–339 (2015)].

    MathSciNet  MATH  Google Scholar 

  17. A. V. Kel’manov and V. I. Khandeev, “An Exact Pseudopolynomial Algorithm for a Problem of the Two-Cluster Partitioning of a Set of Vectors,” Diskretn. Anal. Issled. Oper. 22 (3), 36–48 (2015) [J. Appl. Indust. Math. 9 (4), 497–502 (2015)].

    MathSciNet  MATH  Google Scholar 

  18. A. V. Kel’manov and V. I. Khandeev, “Fully Polynomial-Time Approximation Scheme for Special Case of a Quadratic Euclidean 2-Clustering Problem,” Zh. Vychisl. Mat. Mat. Fiz. 56 (2), 145–153 (2016).

    Google Scholar 

  19. D. Aloise, A. Deshpande, P. Hansen, and P. Popat, “NP-Hardness of Euclidean Sum-of-Squares Clustering,” Mach. Learn. 75 (2), 245–248 (2009).

    Article  Google Scholar 

  20. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, New York, 2006).

    MATH  Google Scholar 

  21. J. A. Carter, E. Agol, et al., “Kepler-36: A Pair of Planets with NeighboringOrbits and DissimilarDensities,” Science 337 (6094), 556–559 (2012).

    Article  Google Scholar 

  22. P. Flach, Machine Learning: The Art and Science of Algorithms That Make Sense of Data (Cambridge Univ. Press, New York, 2012).

    Book  MATH  Google Scholar 

  23. E. Kh. Gimadi, A. V. Kel’manov, M. A. Kel’manova, and S. A. Khamidullin, “A Posteriori Detecting a Quasiperiodic Fragment in a Numerical Sequence,” Pattern Recognit. Image Anal. 18 (1), 30–42 (2008).

    Article  MATH  Google Scholar 

  24. A. K. Jain, “Data Clustering: 50 Years beyond K-Means,” Pattern Recognit. Lett. 31 (8), 651–666 (2010).

    Article  Google Scholar 

  25. G. James, D. Witten, T. Hastie, and D. Tibshirani, An Introduction to Statistical Learning: With Applications in R (Springer, New York, 2013).

    Book  MATH  Google Scholar 

  26. A. V. Kel’manov and B. Jeon, “A Posteriori Joint Detection and Discrimination of Pulses in a Quasiperiodic Pulse Train,” IEEE Trans. Signal Process. 52 (3), 645–656 (2004).

    Article  MathSciNet  Google Scholar 

  27. C. Steger, M. Ulrich, and C. Wiedemann, Machine Vision Algorithms and Applications (Wiley-VCH, Berlin, 2007).

    Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    A. V. Kel’manov, S. A. Khamidullin & V. I. Khandeev

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    A. V. Kel’manov

Authors
  1. A. V. Kel’manov
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  2. S. A. Khamidullin
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  3. V. I. Khandeev
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Correspondence to A. V. Kel’manov.

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Original Russian Text © A.V. Kel’manov, S.A. Khamidullin, V.I. Khandeev, 2016, published in Diskretnyi Analiz i Issledovanie Operatsii, 2016, Vol. 23, No. 2, pp. 21–40.

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Kel’manov, A.V., Khamidullin, S.A. & Khandeev, V.I. A fully polynomial-time approximation scheme for a sequence 2-cluster partitioning problem. J. Appl. Ind. Math. 10, 209–219 (2016). https://doi.org/10.1134/S199047891602006X

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  • DOI: https://doi.org/10.1134/S199047891602006X

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