## Minimum diameter color-spanning sets revisited.(English)Zbl 1506.90231

Summary: We address point sets in the $$d$$-dimensional space, in which every point is colored with at least one color out of the set $$C$$. A Color-Spanning Set or Rainbow Set is a set of points that covers all colors of $$C$$. The diameter of a set is the maximum distance between two points in the set. In this paper we answer some open questions about Minimum Diameter Color-Spanning Sets in $$d$$-dimensional space, which were studied by R. Fleischer and X. Xu [Lect. Notes Comput. Sci. 6213, 285–292 (2010; Zbl 1288.68115)], and extend this concept to general graphs. We show that the problem is W[1] hard for parameter $$|C|$$ and not in PTAS if the number of dimensions is part of the input. Furthermore we demonstrate the membership in W[2]. Most importantly we present two exact solution methods, which both can also be used for the Largest Closest Color-Spanning Set Problem and compare them in an experimental evaluation. For general graphs we show that Minimum Diameter Color-Spanning Set Problem is NP-hard and W[1]-hard for parameter $$| C |$$ but give a polynomial time algorithm for trees. In addition we develop a polynomial time 2-approximation algorithm for general graphs and prove that this problem admits no better approximation factor, unless $$P = N P$$.

### MSC:

 90C27 Combinatorial optimization 05C62 Graph representations (geometric and intersection representations, etc.) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68W25 Approximation algorithms

Zbl 1288.68115

CPLEX
Full Text:

### References:

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