An approximation scheme for a problem of finding a subsequence
A.V. Kelmanov^{1,2}, S.M. Romanchenko^{1}, S.A. Khamidullin^{1}
^{1}Sobolev Institute of Mathematics SB RAS, 4 Acad. Koptyug avenue, Novosibirsk, Russia, 630090 ^{2}Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: последовательность, евклидово пространство, минимум суммы квадратов расстояний, трудность, FPTAS, euclidean space, sequence, minimum sum of squared distances, hardness, FPTAS
Abstract
We consider a strongly 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 polynomialtime approximation scheme (FPTAS) when the space dimension is bounded by a constant.
