S. A. Khamidullin

A posteriori detecting a quasiperiodic fragment in a numerical sequence

Combinatorial approach to solving the problem of detection of an unknown quasi-periodic fragment in a noisy numerical sequence is considered. The problem is analyzed under the following conditions: (1) the number of repeats is known; (2) the number of the sequence term corresponding to the starting instant of the fragment is a deterministic (non-random) value; and (3) the observed sequence is corrupted by additive Gaussian uncorrelated noise. It is demonstarted that the problem under consideration consists in testing the set of composite hypotheses on the mean of a random Gaussian vector. It is shown that the search for a maximum likelihood hypothesis can be reduced to the search for a maximum of an auxiliary objective function. It is proved that the problem of maximization of this function is NP-hard in the general case. An approximate polynomial algorithm for solving the problem is proposed. To improve the approximation, an algorithm of local search is proposed. Numerical simulation showed reasonable results from the applied point of view.

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.

An approximation polynomial-time algorithm for a sequence bi-clustering 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 the minimal sum of the squared distances from the elements of the clusters to the centers of the clusters. The center of one of the clusters is to be optimized and is determined as the mean value over all vectors in this cluster. The center of the other cluster is fixed at the origin. Moreover, the partition is such that the difference between the indices of two successive vectors in the first cluster is bounded above and below by prescribed constants. A 2-approximation polynomial-time algorithm is proposed for this problem.

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.

A fully polynomial-time approximation scheme for a sequence 2-cluster partitioning problem

We consider a strongly NP-hard problem of partitioning a finite sequence of points in Euclidean space into the two clustersminimizing the sum over both clusters of intra-cluster sums of squared distances from the clusters elements to their centers. The sizes of the clusters are fixed. The centroid of the first cluster is defined as the mean value of all vectors in the cluster, and the center of the second cluster is given in advance and equals 0. Additionally, the partition must satisfy the restriction that for all vectors in the first cluster the difference between the indices of two consequent points from this cluster is bounded from below and above by some given constants.We present a fully polynomial-time approximation scheme for the case of fixed space dimension.

An Approximation Scheme for the Problem of Finding a Subsequence

We consider a strongly NP-hard Euclidean problem of finding a subsequence in a finite sequence. The criterion of the solution is the 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 polynomial-time approximation scheme (FPTAS) when the space dimension is bounded by a constant.

Exact pseudopolynomial algorithm for one sequence partitioning problem

We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in a Euclidean space into two clusters of given size with the criterion of minimizing the total sum of square distances from cluster elements to their centers. The center of the first cluster is subject to optimization, defined by the mean value of all vectors in this cluster. The center of the second cluster is fixed at the origin. The partition is subject to the following condition: the difference between indices of two subsequent vectors included in the first cluster is bounded from above and below by given constants. We propose an exact pseudopolynomial algorithm for the case of a problem where the dimension of the space is fixed, and components of input vectors are integer-valued.

An algorithm for recognition of a vector alphabet generating a sequence with a quasi-periodic structure

The problem of noise-proof a posteriori (off-line) recognition is considered for a vector alphabet generating sequences that include quasi-periodically alternating vector-fragments coinciding with elements of this alphabet. One variant of this problem is reduced to the discrete optimization problem under investigation. The exact polynomial solving the algorithm for the reduced problem is proved. This algorithm guarantees the maximum likelihood problem-solving in the case when the noise is additive and it is a Gaussian sequence of independent random values having identical distribution, whereas the number of alternating vector-fragments is unknown. The algorithm suggested has essentially smaller computational complexity in comparison with known one.

A Randomized Algorithm for 2-Partition of a Sequence

In the paper we consider one strongly NP-hard problem of partitioning a finite Euclidean sequence into two clusters minimizing the sum over both clusters of intracluster sum of squared distances from clusters elements to their centers. The cardinalities of clusters are assumed to be given. The center of the first cluster is unknown and is defined as the mean value of all points in the cluster. The center of the second one is the origin. Additionally, the difference between the indexes of two consequent points from the first cluster is bounded from below and above by some constants. A randomized algorithm for the problem is proposed. For an established parameter value, given a relative error \(\varepsilon > 0\) and fixed \(\gamma \in (0, 1)\), this algorithm allows to find a \((1 + \varepsilon )\)-approximate solution of the problem with a probability of at least \(1 - \gamma \) in polynomial time. The conditions are established under which the algorithm is polynomial and asymptotically exact.

Approximation algorithm for the problem of partitioning a sequence into clusters

We consider the problem of partitioning a finite sequence of Euclidean points into a given number of clusters (subsequences) using the criterion of the minimal sum (over all clusters) of intercluster sums of squared distances from the elements of the clusters to their centers. It is assumed that the center of one of the desired clusters is at the origin, while the center of each of the other clusters is unknown and determined as the mean value over all elements in this cluster. Additionally, the partition obeys two structural constraints on the indices of sequence elements contained in the clusters with unknown centers: (1) the concatenation of the indices of elements in these clusters is an increasing sequence, and (2) the difference between an index and the preceding one is bounded above and below by prescribed constants. It is shown that this problem is strongly NP-hard. A 2-approximation algorithm is constructed that is polynomial-time for a fixed number of clusters.