A. V. Pyatkin

Complexity of Certain Problems of Searching for Subsets of Vectors and Cluster Analysis

The discrete extremal problems to which certain problems of searching for subsets of vectors and cluster analysis are reduced are proved to be NP-complete.

The problem of finding a subset of vectors with the maximum total weight

The NP-hardness is proved for the discrete optimization problems connected with maximizing the total weight of a subset of a finite set of vectors in Euclidean space, i.e., the Euclidean norm of the sum of the members. Two approximation algorithms are suggested, and the bounds for the relative error and time complexity are obtained. We give a polynomial approximation scheme in the case of a fixed dimension and distinguished a subclass of problems solvable in pseudopolynomial time. The results obtained are useful for solving the problem of choice of a fixed number of subsequences from a numerical sequence with a given number of quasiperiodical repetitions of the subsequence.

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.

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.

On polynomial solvability of some problems of a vector subset choice in a Euclidean space of fixed dimension

The problems under study are connected with the choice of a vector subset from a given finite set of vectors in the Euclidean space ℝk. The sum norm and averaged square of the sumnorm are considered as the target functions (to be maximized). The optimal combinatorial algorithms with time complexity O(k2n2k) are developed for these problems. Thus, the polynomial solvability of these problems is proved for k fixed.

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.

New formula for the sum number for the complete bipartite graphs

Abstract Given a graph G=(V,E) , a labelling is a function f : V→Z + which has different values on different vertices of G . Graph G is a sum graph if there exists a labelling f : V→Z + such that for every pair of distinct vertices u,v∈V , there is an edge uv∈E if and only if there exists a vertex w∈V with f(w)=f(u)+f(v) . It is clear that every sum graph has at least one isolated vertex. The sum number σ(G) of the graph G is the least number of isolated vertices one must add to G to turn it into a sum graph. It was stated by Hartsfield and Smyth (in: R. Rees (Ed.), Graphs, Matrices and Designs, Marcel Dekker, New York, 1993, pp. 205) that for the complete bipartite graphs K m,n where m⩾n⩾2 the sum number is σ(K m,n )=⌈(3n+m−3)/2⌉ . Unfortunately, this formula is wrong when m⩾3n . The new construction given in this paper shows that σ(K m,n ) in this case is much smaller. The new formula for σ(K m,n ) is proved.

On the complexity of some quadratic Euclidean 2-clustering problems

Some problems of partitioning a finite set of points of Euclidean space into two clusters are considered. In these problems, the following criteria are minimized: (1) the sum over both clusters of the sums of squared pairwise distances between the elements of the cluster and (2) the sum of the (multiplied by the cardinalities of the clusters) sums of squared distances from the elements of the cluster to its geometric center, where the geometric center (or centroid) of a cluster is defined as the mean value of the elements in that cluster. Additionally, another problem close to (2) is considered, where the desired center of one of the clusters is given as input, while the center of the other cluster is unknown (is the variable to be optimized) as in problem (2). Two variants of the problems are analyzed, in which the cardinalities of the clusters are (1) parts of the input or (2) optimization variables. It is proved that all the considered problems are strongly NP-hard and that, in general, there is no fully polynomial-time approximation scheme for them (unless P = NP).

NP-hardness of some Quadratic Euclidean 2-clustering problems

Some problems of partitioning a finite set of points of Euclidean space into two clusters are considered. In these problems, the following criteria are minimized: (1) the sum over both clusters of the sums of squared pairwise distances between the elements of the cluster and (2) the sum of the (multiplied by the cardinalities of the clusters) sums of squared distances from the elements of the cluster to its geometric center. Additionally, another problem close to (2) is considered, in which the desired center of one of the clusters is given as input, while the center of the other cluster is unknown (is the variable to be optimized) as in problem (2). Two variants of the problems are analyzed, in which the cardinalities of the clusters are parts of the input or optimization variables. It is proved that all the considered problems are strongly NP-hard and that, in general, there is no fully polynomial-time approximation scheme for them (unless P = NP).

2-Approximation algorithm for finding a clique with minimum weight of vertices and edges

The problem of finding a minimum clique (with respect to the total weight of its vertices and edges) of fixed size in a complete undirected weighted graph is considered along with some of its important subclasses. Approximability issues are analyzed. The inapproximability of the problem is proved for the general case. A 2-approximation efficient algorithm with time complexity O(n2) is suggested for the cases when vertex weights are nonnegative and edge weights either satisfy the triangle inequality or are squared pairwise distances for some point configuration of Euclidean space.