Geometric Systems of Disjoint Representatives

  • Jiří Fiala
  • Jan Kratochvíl
  • Andrzej Proskurowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2528)


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.


Unit Ball Conjunctive Normal Form Truth Assignment Unit Grid Planar Drawing 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jiří Fiala
    • 1
  • Jan Kratochvíl
    • 1
  • Andrzej Proskurowski
    • 2
  1. 1.Institute for Theoretical Computer Science and Department of Applied MathematicsCharles UniversityPrague
  2. 2.Department of Computer and Information ScienceUniversity of OregonEugene

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