J-means and I-means for minimum sum-of-squares clustering on networks

Given a graph, the Edge minimum sum-of-squares clustering problem requires finding p prototypes (cluster centres) by minimizing the sum of their squared distances from a set of vertices to their nearest prototype, where a prototype can be either a vertex or an inner point of an edge. In this paper we have implemented Variable neighborhood search based heuristic for solving it. We consider three different local search procedures, K-means, J-means, and a new I-means heuristic. Experimental results indicate that the implemented VNS-based heuristic produces the best known results in the literature.     

Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem

The strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters of given sizes (cardinalities) minimizing the sum (over both clusters) of the intracluster sums of squared distances from the elements of the clusters to their centers is considered. It is assumed that the center of one of the sought clusters is specified at the desired (arbitrary) point of space (without loss of generality, at the origin), while the center of the other one is unknown and determined as the mean value over all elements of this cluster. It is shown that unless P = NP, there is no fully polynomial-time approximation scheme for this problem, and such a scheme is substantiated in the case of a fixed space dimension.     

Pseudopolynomial time solvability of a quadratic Euclidean problem of finding a family of disjoint subsets

In this paper, a strongly NP-hard problem of finding a family of disjoint subsets with given cardinalities in a finite set of points from a Euclidean space is considered. Minimization of the sum over all required subsets of the sum of the squared distances from the elements of these subsets to their geometric centers is used as the search criterion. It is proved that if the coordinates of the input points are integer and the space dimension and the number of required subsets are fixed (i.e., bounded by some constants), the problem is a pseudopolynomial time solvable one.     

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.     

Exact pseudopolynomial algorithms for a balanced 2-clustering problem

We consider the strongly NP-hard problem of partitioning a set of Euclidean points into two clusters so as to minimize the sum (over both clusters) of the weighted sum of the squared intracluster distances from the elements of the clusters to their centers. The weights of sums are the sizes of the clusters. The center of one cluster is given as input, while the center of the other cluster is unknown and determined as the average value over all points in the cluster (as the geometric center). Two variants of the problems are analyzed in which the cluster sizes are either given or unknown. We present and prove some exact pseudopolynomial algorithms in the case of integer components of the input points and fixed space dimension.     

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).     

On Improved Bounds for Bounded Degree Spanning Trees for Points in Arbitrary Dimension

Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.559 times the weight of the minimum spanning tree. We also prove that there is a set of points such that no spanning tree of maximal degree 3 exists that has this ratio be less than 1.447. Our central result is based on the proof of the following claim: Given n points in Euclidean space with one special point v, there exists a Hamiltonian path with an endpoint at v that is at most 1.559 times longer than the sum of the distances of the points to v. These proofs also lead to a way to find the tree in linear time given the minimal spanning tree.     

Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center

A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed.     

NP-Hardness of Quadratic Euclidean 1-Mean and 1-Median 2-Clustering Problem with Constraints on the Cluster Sizes

Doklady Mathematics - We consider the problem of clustering a finite set of N points in d-dimensional Euclidean space into two clusters minimizing the sum (over both clusters) of the intracluster...     

Book Reviews

This paper modifies the usual meaning of regression from ‘minimizing the sum of squared distances’ to ‘minimizing the sum of square perpendicular distances’. With this modified definition, the best‐fit plane, circle, and sphere may be meaningfully considered.

These regression problems are motivated by the current development of software to control robotized coordinate measuring machines (CMMs) used to perform quality assurance work. A brief outline of CMMs is given and the application of the modified regression definition to a plane, circle and sphere is illustrated. In §3 the equation of the best fitting plane is calculated for various sets of data points.     

Solving the uncapacitated multi-facility Weber problem by vector quantization and self-organizing maps

The uncapacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane and allocating the demands of n customers to these facilities with the minimum total transportation cost. This is a non-convex optimization problem and difficult to solve exactly. As a consequence, efficient and accurate heuristic solution procedures are needed. The problem has different types based on the distance function used to model the distance between the facilities and customers. We concentrate on the rectilinear and Euclidean problems and propose new vector quantization and self-organizing map algorithms. They incorporate the properties of the distance function to their update rules, which makes them different from the existing two neural network methods that use rather ad hoc squared Euclidean metric in their updates even though the problem is originally stated in terms of the rectilinear and Euclidean distances. Computational results on benchmark instances indicate that the new methods are better than the existing ones, both in terms of the solution quality and computation time.     

A global optimization procedure for the location of a median line in the three-dimensional space

A global optimization procedure is proposed to find a line in the Euclidean three-dimensional space which minimizes the sum of distances to a given finite set of three-dimensional data points.Although we are using similar techniques as for location problems in two dimensions, it is shown that the problem becomes much harder to solve. However, a problem parameterization as well as lower bounds are suggested whereby we succeeded in solving medium-size instances in a reasonable amount of computing time.     

A clustering approach to the planar hub location problem

The problem tackled in this paper is as follows: consider a set ofn interacting points in a two-dimensional space. The levels of interactions between the observations are given exogenously. It is required to cluster then observations intop groups, so that the sum of squared deviations from the cluster means is as small as possible. Further, assume that the cluster means are adjusted to reflect the interaction between the entities. (It is this latter consideration which makes the problem interesting.) A useful property of the problem is that the use of a squared distance term yields a linear system of equations for the coordinates of the cluster centroids. These equations are derived and solved repeatedly for a given set of cluster allocations. A sequential reallocation of the observations between the clusters is then performed. One possible application of this problem is to the planar hub location problem, where the interacting observations are a system of cities and the interaction effects represent the levels of flow or movement between the entities. The planar hub location problem has been limited so far to problems with fewer than 100 nodes. The use of the squared distance formulation, and a powerful supercomputer (Cray Y-MP) has enabled quick solution of large systems with 250 points and four groups. The paper includes both small illustrative examples and computational results using systems with up to 500 observations and 9 clusters.     

Linear separation and approximation by minimizing the sum of concave functions of distances

One recently proposed criterion to separate two data sets in Classification is to use a hyperplane that minimizes the sum of distances to it from all the misclassified data points, where misclassification means lying on the wrong side of the hyperplane, or rather in the wrong halfspace. In this paper we study an extension of this problem: we seek the hyperplane minimizing the sum of concave nondecreasing functions of the distances of misclassified points to it. It is shown that an optimal hyperplane exists containing at least $d$ affinely independent points. This extends the result known for the minimization of the sum of distances, and enables to use combinatorial local-search heuristics for this problem. As a corollary, the same result is obtained for the approximation problem in which a hyperplane minimizing the sum of concave nondecreasing functions of the distances from a set of data points is sought.     

Sphere-separable partitions of multi-parameter elements

We show the optimality of sphere-separable partitions for problems where n vectors in d-dimensional space are to be partitioned into p categories to minimize a cost function which is dependent in the sum of the vectors in each category; the sum of the squares of their Euclidean norms; and the number of elements in each category. We further show that the number of these partitions is polynomial in n. These results broaden the class of partition problems for which an optimal solution is guaranteed within a prescribed set whose size is polynomially bounded in n. Applications of the results are demonstrated through examples.     

Local convergence in a generalized Fermat-Weber problem

In the Fermat-Weber problem, the location of a source point in ^N is sought which minimizes the sum of weighted Euclidean distances to a set of destinations. A classical iterative algorithm known as the Weiszfeld procedure is used to find the optimal location. Kuhn proves global convergence except for a denumerable set of starting points, while Katz provides local convergence results for this algorithm. In this paper, we consider a generalized version of the Fermat-Weber problem, where distances are measured by anl _p norm and the parameterp takes on a value in the closed interval [1, 2]. This permits the choice of a continuum of distance measures from rectangular (p=1) to Euclidean (p=2). An extended version of the Weiszfeld procedure is presented and local convergence results obtained for the generalized problem. Linear asymptotic convergence rates are typically observed. However, in special cases where the optimal solution occurs at a singular point of the iteration functions, this rate can vary from sublinear to quadratic. It is also shown that for sufficiently large values ofp exceeding 2, convergence of the Weiszfeld algorithm will not occur in general.     

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 stochastic generation of ultrametrics in high-dimensional Euclidean spaces

We present a proof of the theorem which states that a matrix of Euclidean distances on a set of specially distributed random points in the n-dimensional Euclidean space R ⁿ converges in probability to an ultrametric matrix as n → ∞. Values of the elements of an ultrametric distance matrix are completely determined by variances of coordinates of random points. Also we present a probabilistic algorithm for generation of finite ultrametric structures of any topology in high-dimensional Euclidean space. Validity of the algorithm is demonstrated by explicit calculations of distance matrices and ultrametricity indexes for various dimensions n.     

An optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications

Given a set of entities associated with points in Euclidean space, minimum sum-of-squares clustering (MSSC) consists in partitioning this set into clusters such that the sum of squared distances from each point to the centroid of its cluster is minimized. A column generation algorithm for MSSC was given by du Merle et?al. in SIAM Journal Scientific Computing 21:1485–1505. The bottleneck of that algorithm is the resolution of the auxiliary problem of finding a column with negative reduced cost. We propose a new way to solve this auxiliary problem based on geometric arguments. This greatly improves the efficiency of the whole algorithm and leads to exact solution of instances with over 2,300 entities, i.e., more than 10 times as much as previously done.     

Properties of Three-Dimensional Median Line Location Models

We consider the problem of locating a line with respect to some existing facilities in 3-dimensional space, such that the sum of weighted distances between the line and the facilities is minimized. Measuring distance using the l _p norm is discussed, along with the special cases of Euclidean and rectangular norms. Heuristic solution procedures for finding a local minimum are outlined.