The following two strongly NP-hard problems are considered. In the first problem, we need to find in the given finite set of points in Euclidean space the subset of largest size. The sum of squared distances between the elements of this subset and its unknown centroid (geometrical center) must not exceed a given value. This value is defined as percentage of the sum of squared distances between the elements of the input set and its centroid. In the second problem, the input is a sequence (not a set) and we have some additional constraints on the indices of the elements of the chosen subsequence. The restriction on the sum of squared distances is the same as in the first problem. Both problems can be treated as data editing problems aimed to find similar elements and removal of extraneous (dissimilar) elements. We propose exact algorithms for the cases of both problems in which the input points have integer-valued coordinates. If the space dimension is bounded by some constant, our algorithms run in a pseudopolynomial time. Some results of numerical experiments illustrating the performance of the algorithms are presented.

中文翻译：

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搜索最大子集和最长子序列的两个整数值问题的精确算法

考虑了以下两个强 NP-hard 问题。在第一个问题中，我们需要在给定的欧几里得空间中的有限点集中找到最大尺寸的子集。此子集的元素与其未知质心（几何中心）之间的平方距离总和不得超过给定值。该值定义为输入集元素与其质心之间距离平方和的百分比。在第二个问题中，输入是一个序列（不是一个集合），我们对所选子序列的元素的索引有一些额外的约束。距离平方和的限制与第一个问题相同。这两个问题都可以被视为数据编辑问题，旨在找到相似的元素并去除无关（不同）的元素。我们为输入点具有整数值坐标的两个问题的情况提出了精确算法。如果空间维度受某个常数的限制，我们的算法在伪多项式时间内运行。给出了说明算法性能的一些数值实验结果。