We study the question of fair clustering under the disparate impact doctrine, where each protected class must have approximately equal representation in every cluster. We formulate the fair clustering problem under both the k-center and the k-median objectives, and show that even with two protected classes the problem is challenging, as the optimum solution can violate common conventions—for in...

The anchored hyperplane location problem is to locate a hyperplane passing through some given points P ⊆ IR and minimizing either the sum of weighted distances (median problem), or the maximum weighted distance (center problem) to some other points Q ⊆ IR. This problem of computational geometry is analyzed by using nonlinear programming techniques. If the distances are measured by a norm, it wi...

This paper concerns the minimax center of a collection linear subspaces. For k-dimensional subspaces an n-dimensional vector space, this can be cast as finding minimum enclosing ball on Grassmann manifold. differing dimension, setting becomes disjoint union Grassmannians rather than single manifold, and problem is no longer well-defined. However, natural geometric maps exist between these manif...

The vortex contribution to the k-string tensions is computed for SU(N) gauge theories. We deduce the surface densities needed to reproduce the sine scaling and the Casimir scaling formulae, recently obtained from numerical simulations on the lattice. We find that such densities need to grow linearly in N , which in turn suggests that the vortex scenario can hardly reproduce the physics of confi...

We study local search algorithms for metric instances of facility location problems: the uncapacitated facility location problem (UFL), as well as uncapacitated versions of the k-median, k-center and kmeans problems. All these problems admit natural local search heuristics: for example, in the UFL problem the natural moves are to open a new facility, close an existing facility, and to swap a cl...

ICGA 91 Intelligent Structural Operators for the k-way Graph Partitioning Problem 0 Intelligent Structural Operators for the k-way Graph Partitioning Problem Gregor von Laszewski Gesellschaft f ur Mathematik und Datenverarbeitung mbH Schlo Birlinghoven, D { 5205 St. Augustin [email protected] NorthEast Parallel Architectures Center Syracuse University 111 College Place Syracuse, NY 13244-4100

This paper considers a multiple-circle detection problem on the basis of given data. The problem is solved by application of the center-based clustering method. For the purpose of searching for a locally optimal partition modeled on the well-known k-means algorithm, the k-closest circles algorithm has been constructed. The method has been illustrated by several numerical examples.

Large-scale clustering of data points in metric spaces is an important problem in mining big data sets. For many applications, we face explicit or implicit size constraints for each cluster which leads to the problem of clustering under capacity constraints or the “balanced clustering” problem. Although the balanced clustering problem has been widely studied, developing a theoretically sound di...

Given a metric (U, d) with |U | = n, a subset S ⊆ U with |S| = l, a non-negative function w : U → <, and an integer k ≤ l, the generalized static k-center and k-median problems ask to pick a subset X ⊆ S with |X| = k so as to minimize maxx∈U d(x,X) and ∑ x∈U d(x,X) · w(x), respectively, where d(x,X) = miny∈X d(x, y). Each point in X is called a center and each point in U is assigned to its clos...

In the k-means problem, we are given a finite set S of points in Rm, and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [7].