## Algorithmic aspects of the maximum colorful arborescence problem.(English)Zbl 1485.68181

Gopal, T. V. (ed.) et al., Theory and applications of models of computation. 14th annual conference, TAMC 2017, Bern, Switzerland, April 20–22, 2017. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 10185, 216-230 (2017).
Summary: Given a vertex-colored arc-weighted directed acyclic graph $$G$$, the Maximum Colorful Subtree problem (or MCS) aims at finding an arborescence of maximum weight in $$G$$, in which no color appears more than once. The problem was originally introduced in [S. Böcker et al., “Towards de novo identification of metabolites by analyzing tandem mass spectra”, Bioinf. 24, No. 16, i49–i55 (2008; doi:10.1093/bioinformatics/btn270)] in the context of de novo identification of metabolites by tandem mass spectrometry. However, a thorough analysis of the initial motivation shows that the formal definition of MCS needs to be amended, since the input graph $$G$$ actually possesses two extra properties, which are so far unexploited. This leads us to introduce in this paper a more precise model that we call Maximum Colorful Arborescence (MCA), and extensively study it in terms of algorithmic complexity. In particular, we show that exploiting the implied color hierarchy of the input graph can lead to polynomial algorithms. We also develop fixed-parameter tractable (FPT) algorithms for the problem, notably using the “dual parameter” $$\ell$$, defined as the number of vertices of $$G$$ which are not kept in the solution.
For the entire collection see [Zbl 1360.68012].

### MSC:

 68R10 Graph theory (including graph drawing) in computer science 05C22 Signed and weighted graphs 68Q25 Analysis of algorithms and problem complexity 68W05 Nonnumerical algorithms 92C40 Biochemistry, molecular biology

### Software:

speedy_colorful_subtrees
Full Text:

### References:

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