Approximation algorithms for some intractable problems of choosing a vector subsequence

We consider some intractable optimization problems of finding a subsequence in a finite sequence of vectors of the Euclidean space. We assume that the sought subsequence contains a fixed number of vectors close to each other under the criterion of the minimum sum of the squares of distances. Moreoveer, this subsequence has to satisfy the following condition: the difference between the indexes of each previous and next vectors of the sought subsequence is bounded with lower and upper constants. Some 2-approximation efficient algorithms for solving these problems are introduced.