Fault-Tolerant Covering Problems in Metric Spaces

In this article, we study some fault-tolerant covering problems in metric spaces. In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space ( X ∪ Y , d ) , a positive integer k that represents the coverage demand of each client, and a constant α ≥ 1 . Each server can host a single ball of arbitrary radius centered on it. Each client x ∈ X needs to be covered by at least k such balls centered on servers. The objective function that we wish to minimize is the sum of the α -th powers of the radii of the balls. We also study some non-trivial generalizations of the MMC, such as (a) the non-uniform MMC , where we allow client-specific demands, and (b) the t - MMC , where we require the number of open servers to be at most some given integer t . We present the first constant approximations for these fault-tolerant covering problems. Our algorithms are based on the following paradigm: for each of the three problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding 1-covering problem, where the coverage demand of each client is 1. The reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for 1-covering, we obtain our results for the MMC and these generalizations.