Abstract
We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of minimum sum-of-squares of distances from the elements of clusters to their centers. We assume that the cardinalities of the clusters are fixed. The center of one cluster has to be optimized and is defined as the average value over all vectors in this cluster. The center of the other cluster lies at the origin. The partition satisfies the condition: the difference of the indices of the next and previous vectors in the first cluster is bounded above and below by two given constants. We propose a 2-approximation polynomial algorithm to solve this problem.
Similar content being viewed by others
References
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)].
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)].
A. V. Kel’manov and S. M. Romanchenko, “An Approximation Algorithm for Solving a Problem of Search for a Vector Subset,” Diskretn. Anal. Issled. Oper. 18(1), 61–69 (2011) [J. Appl. Indust. Math. 6 (1), 90–96 (2012)].
A. V. Kel’manov and S. A. Khamidullin, “Posteriori Detection of a Given Number of Identical Subsequences in a Quasiperiodic Sequence,” Zh. Vychisl. Mat. Mat. Fiz. 41(5), 807–820 (2001) [Comput. Math. Math. Physics. 41 (5), 762–774 (2001)].
D. Aloise, A. Deshpande, P. Hansen, and P. Popat, “NP-Hardness of Euclidean Sum-of-Squares Clustering,” G-2008-33 (Les Cahiers du GERAD, 2008).
K. Anil and K. Jain, “Data Clustering: 50 Years Beyond k-Means,” Pattern Recognit. Lett. 31, 651–666 (2010).
M. R. Garey and D. S. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).
T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, New York, 2001).
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Kel’manov, S.A. Khamidullin, 2014, published in Diskretnyi Analiz i Issledovanie Operatsii, 2014, Vol. 21, No. 1, pp. 53–66.
Rights and permissions
About this article
Cite this article
Kel’manov, A.V., Khamidullin, S.A. An approximating polynomial algorithm for a sequence partitioning problem. J. Appl. Ind. Math. 8, 236–244 (2014). https://doi.org/10.1134/S1990478914020100
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478914020100