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In the Metric Capacitated Covering (MCC) problem, given a set of balls $mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $mathcal{B}subseteq mathcal{B}$ and an assignment of the points in $P$ to the balls in $mathcal{B}$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(log |P|)$ even with $beta < 3$ factor expansion of the balls. Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] showed that one can obtain an $O(1)$-approximation for the problem with $6.47$ factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound $3$ and the upper bound $6.47$. In this current work, we show that it is possible to obtain an $O(1)$-approximation with only $4.24$ factor expansion of the balls. We also show a similar upper bound of $5$ for a more generalized version of MCC for which the best previously known bound was $9$.
In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each cente
In this paper, we study the lower- and upper-bounded covering (LUC) problem, where we are given a set $P$ of $n$ points, a collection $mathcal{B}$ of balls, and parameters $L$ and $U$. The goal is to find a minimum-sized subset $mathcal{B}subseteq ma
A covering code is a subset $mathcal{C} subseteq {0,1}^n$ with the property that any $z in {0,1}^n$ is close to some $c in mathcal{C}$ in Hamming distance. For every $epsilon,delta>0$, we show a construction of a family of codes with relative coverin
In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clients) in an arbitrary metric space $(X cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $alpha geq 1$.
The $k$-Facility Location problem is a generalization of the classical problems $k$-Median and Facility Location. The goal is to select a subset of at most $k$ facilities that minimizes the total cost of opened facilities and established connections