Capacitated Covering Problems in Geometric Spaces

In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B\u27 subseteq B of balls and assign each point in P to some ball in B\u27 that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B\u27. We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only (1+epsilon) factor expansion is sufficient for any epsilon > 0, with the approximation factor being a polynomial in 1/epsilon. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems

A Primal-Dual Based Power Control Approach for Capacitated Edge Servers

The intensity of radio waves decays rapidly with increasing propagation distance, and an edge server's antenna needs more power to form a larger signal coverage area. Therefore, the power of the edge server should be controlled to reduce energy consumption. In addition, edge servers with capacitated resources provide services for only a limited number of users to ensure the quality of service (QoS). We set the signal transmission power for the antenna of each edge server and formed a signal disk, ensuring that all users were covered by the edge server signal and minimizing the total power of the system. This scenario is a typical geometric set covering problem, and even simple cases without capacity limits are NP-hard problems. In this paper, we propose a primal-dual-based algorithm and obtain an $m$-approximation result. We compare our algorithm with two other algorithms through simulation experiments. The results show that our algorithm obtains a result close to the optimal value in polynomial time