Fault-Tolerant Covering Problems in Metric Spaces

Abstract

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 $$(X \cup Y, d)$$ , a positive integer k that represents the coverage demand of each client, and a constant $$\alpha \ge 1$$ . Each server can host a single ball of arbitrary radius centered on it. Each client $$x \in X$$ 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 $$\alpha $$ -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.