On Complexity of Searching a Subset of Vectors with Shortest Average Under a Cardinality Restriction
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In this paper, we study the computational complexity of the following subset search problem in a set of vectors. Given a set of N Euclidean q-dimensional vectors and an integer M, choose a subset of at least M vectors minimizing the Euclidean norm of the arithmetic mean of chosen vectors. This problem is induced, in particular, by a problem of clustering a set of points into two clusters where one of the clusters consists of points with a mean close to a given point. Without loss of generality the given point may be assumed to be the origin.
We show that the considered problem is NP-hard in the strong sense and it does not admit any approximation algorithm with guaranteed performance, unless P = NP. An exact algorithm with pseudo-polynomial time complexity is proposed for the special case of the problem, where the dimension q of the space is bounded from above by a constant and the input data are integer.
KeywordsVectors sum Subset selection Euclidean norm NP-hardness Pseudo-polymonial time
This research is supported by RFBR, projects 15-01-00462, 16-01-00740 and 15-01-00976.
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