# Geometric Systems of Disjoint Representatives

## Abstract

Consider a finite collection of subsets of a metric space and ask for a system of representatives which are pairwise at a distance at least *q*, where *q* is a parameter of the problem. In discrete spaces this generalizes the well known problem of distinct representatives, while in Euclidean metrics the problem reduces to finding a system of disjoint balls. This problem is closely related to practical applications like scheduling or map labeling. We characterize the computational complexity of this geometric problem for the cases of *L* _{1} and *L* _{2} metrics and dimensions *d* = 1, 2. We show that for *d* = 1 the problem can be solved in polynomial time, while for *d* = 2 we prove that it is *NP*-hard. Our *NP*-hardness proof can be adjusted also for higher dimensions.

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