A. V. Kel’manov

An approximation algorithm for solving a problem of search for a vector subset

One of the problems in data analysis was earlier reduced to a specific NP-hard optimization problem of finding in a given vector set in the Euclidean space a subset of a given cardinality such that the subset consists of the vectors that are “close” to each other by the criterion of the minimum sum of squared distances. In the paper an efficient 2-approximation algorithm is proposed for solving this problem.

NP-completeness of some problems of choosing a vector subset

The NP-completeness is proved of some problems of choosing a Euclidean vector subset. One of the data analysis problems is reduced to these problems. The required subset is assumed to have a fixed cardinality and include the vectors that are “close” to each other by the criterium of the minimum sum of squares of distances.

An FPTAS for a vector subset search problem

Under study is a strongly NP-hard problem of finding a subset of a given size of a finite set of vectors in Euclidean space which minimizes the sum of squared distances from the elements of this subset to its center. The center of the subset is defined as the average vector calculated with all subset elements. It is proved that, unless P=NP, in the general case of the problem there is no fully polynomial time approximation scheme (FPTAS). Such a scheme is provided in the case when the dimension of the space is fixed.

On complexity of some problems of cluster analysis of vector sequences

NP-completeness of two clustering (partition) problems is proved for a finite sequence of Euclidean vectors. In the optimization versions of both problems it is required to partition the elements of the sequence into a fixed number of clusters minimizing the sum of squares of the distances from the cluster elements to their centers. In the first problem the sizes of clusters are the part of input, while in the second they are unknown (they are the variables for optimization). Except for the center of one (special) cluster, the center of each cluster is the mean value of all vectors contained in it. The center of the special cluster is zero. Also, the partition must satisfy the following condition: The difference between the indices of two consecutive vectors in every nonspecial cluster is bounded below and above by two given constants.

An approximation algorithm for solving a problem of cluster analysis

The authors provide some 2-approximation algorithm for an intractable problem to which one can reduce the problem of partitioning a vector set in Euclidean space into the two subsets (clusters) having the minimum sum of distance squares.

On a Version of the Problem of Choosing a Vector Subset

The NP-completeness is proved of the problem of choosing some subset of “similar” vectors. One of the variants of the a posteriori (off-line) noise-proof detection problem of an unknown repeating vector in a numeric sequence can be reduced to this problem in the case of additive noise. An approximation polynomial algorithm with a guaranteed performance bound is suggested for this problem in the case of a fixed space dimension.

An exact pseudopolynomial algorithm for a problem of the two-cluster partitioning of a set of vectors

We consider the strongly NP-hard problem of partitioning a set of Euclidean vectors into two clusters of given sizes so as to minimize the sum of the squared distances from the elements of the clusters to their centers. It is assumed that the center of one of the clusters is unknown and determined as the average value over all vectors in the cluster. The center of the other cluster is the origin.We prove that, for a fixed dimension of the space, the problem is solvable in polynomial time. We also present and justify an exact pseudopolynomial algorithm in the case of integer components of the vectors.

A 2-approximation polynomial algorithm for a clustering problem

A 2-approximation algorithm is presented for some NP-hard data analysis problem that consists in partitioning a set of Euclidean vectors into two subsets (clusters) under the criterion of minimum sum-of-squares of distances from the elements of clusters to their centers. The center of the first cluster is the average value of vectors in the cluster, and the center of the second one is the origin.

An Approximating Polynomial Algorithm for a Sequence Partitioning Problem

We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of minimum sum-of-squares of distances from the elements of clusters to their centers. We assume that the cardinalities of the clusters are fixed. The center of one cluster has to be optimized and is defined as the average value over all vectors in this cluster. The center of the other cluster lies at the origin. The partition satisfies the condition: the difference of the indices of the next and previous vectors in the first cluster is bounded above and below by two given constants. We propose a 2-approximation polynomial algorithm to solve this problem.

On the algorithmic complexity of a problem in cluster analysis

We prove that the MSSC problem (the problem of clustering the set of the vectors in the Euclidean space which minimizes the sum of squares) is NP-complete in the case when the dimension of the space is an input parameter of the problem, while the number of clusters is not an input parameter.