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.
The authors acknowledge support of joint Czech U.S. grants KONTAKT ME338 and NSF-INT-9802416 during visits of the first two authors to Eugene, OR, and of the third author to Prague.
Research partially supported by Czech Research grant GAUK 158/99.
Supported in part by the grant NSF-ANI-9977524. Supported by the Ministry of Education of the Czech Republic as project LN00A056.
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© 2002 Springer-Verlag Berlin Heidelberg
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Fiala, J., Kratochvíl, J., Proskurowski, A. (2002). Geometric Systems of Disjoint Representatives. In: Goodrich, M.T., Kobourov, S.G. (eds) Graph Drawing. GD 2002. Lecture Notes in Computer Science, vol 2528. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36151-0_11
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DOI: https://doi.org/10.1007/3-540-36151-0_11
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