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**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 |

### Keywords:

minimum diameter; color-spanning sets; multicolored clique; W-hierarchy; approximation algorithms; IQP formulation### Citations:

Zbl 1288.68115### Software:

CPLEX
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XMLCite

\textit{J. Pruente}, Discrete Optim. 34, Article ID 100550, 15 p. (2019; Zbl 1506.90231)

Full Text:
DOI

### References:

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