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We present filling as a new type of spatial subdivision problem that is related to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most…
We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning…
The paper outlines the principles of construction of a broad class of hierarchical aggregation algorithms of cluster analysis, essentially based on minimum distance mergers, which are derived from the general bi-partial objective function.…
We study the problem of minimizing a multivariate polynomial function over the unit hypercube. By representing the polynomial through a hypergraph and exploiting its sparsity structure, we establish a new sufficient condition under which…
We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding $\varphi:G\rightarrow M$ of a graph $G$ into a 2-manifold $M$ maps the vertices in $V(G)$ to distinct points and the…
We consider the problem of finding a subgraph of a given graph minimizing the sum of given functions at vertices evaluated at their subgraph degrees. While the problem is NP-hard already for bipartite graphs when the functions are convex on…
In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region, or…
We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity…
As individual traffic and public transport in cities are changing, city authorities need to analyze urban geospatial data to improve transportation and infrastructure. To that end, they highly rely on spatial aggregation queries that…
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…
The optimal circle coverage problem aims to find a configuration of circles that maximizes the covered area within a given region. Although theoretical optimal solutions exist for simple cases, the problem's NP-hard characteristic makes the…
Biclustering, also called co-clustering, block clustering, or two-way clustering, involves the simultaneous clustering of both the rows and columns of a data matrix into distinct groups, such that the rows and columns within a group display…
We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…
Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. In the abstract form of the problem, where no noise or other corruptions are present, the data are assumed to lie in general position…
We are interested in multilayer graph clustering, which aims at dividing the graph nodes into categories or communities. To do so, we propose to learn a clustering-friendly embedding of the graph nodes by solving an optimization problem…
In this paper, we propose a topology optimization (TO) framework where the design is parameterized by a set of convex polygons. Extending feature mapping methods in TO, the representation allows for direct extraction of the geometry. In…
This article investigates a distributed aggregative optimization problem subject to coupled affine inequality constraints, in which local objective functions depend not only on their own decision variables but also on an aggregation of all…
Community-based graph clustering is one of the most popular topics in the analysis of complex social networks. This type of clustering involves grouping vertices that are considered to share more connections, whereas vertices in different…
The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low…
In this article, we consider a collection of geometric problems involving points colored by two colors (red and blue), referred to as bichromatic problems. The motivation behind studying these problems is two fold; (i) these problems appear…