A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Applied Mathematics
Map labeling and its generalizations
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Hall's theorem for hypergraphs
Journal of Graph Theory
A fast algorithm for single processor scheduling
SFCS '78 Proceedings of the 19th Annual Symposium on Foundations of Computer Science
Systems of distant representatives
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Approximation algorithms for spreading points
Journal of Algorithms
Systems of distant representatives
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Approximation algorithms for spreading points
WAOA'04 Proceedings of the Second international conference on Approximation and Online Algorithms
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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 L1 and L2 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.