On strong chains of sets and functions

Shelah has shown that there are no chains of length $\omega_3$ increasing modulo finite in ${}^{\omega_2}\omega_2$. We improve this result to sets. That is, we show that there are no chains of length $\omega_3$ in $[\omega_2]^{\aleph_2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length $\omega_2$ increasing modulo finite in $[\omega_1]^{\aleph_1}$ as well as in ${}^{\omega_1}\omega_1$. More generally, we study the depth of function spaces ${}^\kappa\mu$ quotiented by the ideal $[\kappa]^{< \theta}$ where $\theta< \kappa$ are infinite cardinals

On Partial Covering For Geometric Set Systems

We study a generalization of the Set Cover problem called the Partial Set Cover in the context of geometric set systems. The input to this problem is a set system (X, R), where X is a set of elements and R is a collection of subsets of X, and an integer k <= |X|. Each set in R has a non-negative weight associated with it. The goal is to cover at least k elements of X by using a minimum-weight collection of sets from R. The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for a certain projection of the set system (X, R). As a corollary of this result, we get improved approximation guarantees for the Partial Set Cover problem for a large class of geometric set systems

A Constant Approximation for Colorful k-Center

In this paper, we consider the colorful k-center problem, which is a generalization of the well-known k-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to find the smallest radius rho, such that with k balls of radius rho, the desired number of points of each color can be covered. We obtain a constant approximation for this problem in the Euclidean plane. We obtain this result by combining a "pseudo-approximation" algorithm that works in any metric space, and an approximation algorithm that works for a special class of instances in the plane. The latter algorithm uses a novel connection to a certain matching problem in graphs

Non-Uniform k-Center and Greedy Clustering

In the Non-Uniform k-Center (NUkC) problem, a generalization of the famous k-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t-NUkC, we assume that the number of distinct radii is equal to t, and we are allowed to use k_i balls of radius r_i, for 1 ≤ i ≤ t. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t-NUkC is not possible if t is unbounded, assuming ≠ NP. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t-NUkC should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture - currently, a constant approximation for 3-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O(1)-approximation for the Non-Uniform k-Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k-center literature, which also demonstrates that the different generalizations of k-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.publishedVersio