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We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary…
We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most…
$k$-means clustering is a well-studied problem due to its wide applicability. Unfortunately, there exist strong theoretical limits on the performance of any algorithm for the $k$-means problem on worst-case inputs. To overcome this barrier,…
Given a complete graph with $n$ vertices and non-negative edge weights, where $n$ is divisible by 3, the maximum weight 3-path packing problem is to find a set of $n/3$ vertex-disjoint 3-paths such that the total weight of the 3-paths in…
The method of Hol\'y, Sokol and \v{C}ern\'y (Applied Soft Computing, 2017, Vol. 60, p. 752-762) clusters objects based on their incidence in a large number of given sets. The idea is to minimize the occurrence of multiple objects from the…
We design new parallel algorithms for clustering in high-dimensional Euclidean spaces. These algorithms run in the Massively Parallel Computation (MPC) model, and are fully scalable, meaning that the local memory in each machine may be…
We study a generalization of the knapsack problem with geometric and vector constraints. The input is a set of rectangular items, each with an associated profit and $d$ nonnegative weights ($d$-dimensional vector), and a square knapsack.…
In the matroid partitioning problem, we are given $k$ matroids $\mathcal{M}_1 = (V, \mathcal{I}_1), \dots , \mathcal{M}_k = (V, \mathcal{I}_k)$ defined over a common ground set $V$ of $n$ elements, and we need to find a partitionable set $S…
The objective of clustering is to discover natural groups in datasets and to identify geometrical structures which might reside there, without assuming any prior knowledge on the characteristics of the data. The problem can be seen as…
We study the optimization version of the set partition problem (where the difference between the partition sums are minimized), which has numerous applications in decision theory literature. While the set partitioning problem is NP-hard and…
Cluster analysis requires many decisions: the clustering method and the implied reference model, the number of clusters and, often, several hyper-parameters and algorithms' tunings. In practice, one produces several partitions, and a final…
Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the EM algorithm. The conventional approach to determining a suitable number of components is to compare…
Comparison of three kind of the clustering and find cost function and loss function and calculate them. Error rate of the clustering methods and how to calculate the error percentage always be one on the important factor for evaluating the…
One of the most useful measures of cluster quality is the modularity of a partition, which measures the difference between the number of the edges joining vertices from the same cluster and the expected number of such edges in a random…
String matching is the problem of finding all the substrings of a text which match a given pattern. It is one of the most investigated problems in computer science, mainly due to its very diverse applications in several fields. Recently,…
Clustering is the problem of separating a set of objects into groups (called clusters) so that objects within the same cluster are more similar to each other than to those in different clusters. Spectral clustering is a now well-known…
Clustering plays a crucial role in computer science, facilitating data analysis and problem-solving across numerous fields. By partitioning large datasets into meaningful groups, clustering reveals hidden structures and relationships within…
The clustering of a data set is one of the core tasks in data analytics. Many clustering algorithms exhibit a strong contrast between a favorable performance in practice and bad theoretical worst-cases. Prime examples are least-squares…
Vectors of data are at the heart of machine learning and data mining. Recently, vector quantization methods have shown great promise in reducing both the time and space costs of operating on vectors. We introduce a vector quantization…
One important tool is the optimal clustering of data into useful categories. Dividing similar objects into a smaller number of clusters is of importance in many applications. These include search engines, monitoring of academic performance,…