Geodesic
An approximation scheme for a~problem of finding a~subsequence
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 4, pp. 379-392

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We consider a strongly NP-hard Euclidean problem of finding a subsequence in a finite sequence. The criterion of the solution is a minimum sum of squared distances from the elements of a sought subsequence to its geometric center (centroid). It is assumed that a sought subsequence contains a given number of elements. In addition, a sought subsequence should satisfy the following condition: the difference between the indices of each previous and subsequent points is bounded with given lower and upper constants. We present an approximation algorithm of solving the problem and prove that it is a fully polynomial-time approximation scheme (FPTAS) when the space dimension is bounded by a constant. 
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     title = {An approximation scheme for a~problem of finding a~subsequence},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
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     url = {http://geodesic.mathdoc.fr/item/SJVM_2017_20_4_a2/}
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A. V. Kelmanov; S. M. Romanchenko; S. A. Khamidullin. An approximation scheme for a~problem of finding a~subsequence. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 4, pp. 379-392. http://geodesic.mathdoc.fr/item/SJVM_2017_20_4_a2/