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On some geometric problems of color-spanning sets. (English) Zbl 1275.90080

Summary: We study several geometric problems of color-spanning sets: given \(n\) points with \(m\) colors in the plane, selecting \(m\) points with \(m\) distinct colors such that some geometric properties of the \(m\) selected points are minimized or maximized. The geometric properties are the maximum diameter, the largest closest pair, the planar smallest minimum spanning tree, the planar largest minimum spanning tree and the planar smallest perimeter convex hull. We propose an \(O(n ^{1+\varepsilon })\) time algorithm for the maximum diameter color-spanning set problem where \(\varepsilon \) could be an arbitrarily small positive constant. Then, we present hardness proofs for the other problems and propose two efficient constant factor approximation algorithms for the planar smallest perimeter color-spanning convex hull problem.

MSC:

90C27 Combinatorial optimization
90C60 Abstract computational complexity for mathematical programming problems
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