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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.
The Non-Uniform $k$-center (NUkC) problem has recently been formulated by Chakrabarty, Goyal and Krishnaswamy [ICALP, 2016] as a generalization of the classical $k$-center clustering problem. In NUkC, given a set of $n$ points $P$ in a metric space a
The quest for colorful components (connected components where each color is associated with at most one vertex) inside a vertex-colored graph has been widely considered in the last ten years. Here we consider two variants, Minimum Colorful Components
For any $epsilon>0$, Laue and Matijevi{c} [CCCG07, IPL08] give a PTAS for finding a $(1+epsilon)$-approximate solution to the $k$-hop MST problem in the Euclidean plane that runs in time $(n/epsilon)^{O(k/epsilon)}$. In this paper, we present an algo
Given a graph $G=(V,E)$ and an integer $k ge 1$, a $k$-hop dominating set $D$ of $G$ is a subset of $V$, such that, for every vertex $v in V$, there exists a node $u in D$ whose hop-distance from $v$ is at most $k$. A $k$-hop dominating set of minimu
Given a data set of size $n$ in $d$-dimensional Euclidean space, the $k$-means problem asks for a set of $k$ points (called centers) so that the sum of the $ell_2^2$-distances between points of a given data set of size $n$ and the set of $k$ centers