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The Maximum Colorful Arborescence problem: how (computationally) hard can it be? (English) Zbl 1477.68224

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 and F. Rasche, “Towards de novo identification of metabolites by analyzing tandem mass spectra”, Bioinform. 24, No. 16, 49–55 (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 should be amended, since the input graph \(G\) actually possesses extra properties, which have been unexploited so far. This leads us to describe in this paper a more precise model that we call Maximum Colorful Arborescence (MCA), which we extensively study in terms of algorithmic complexity. In particular, we show that exploiting the implied Color Hierarchy Graph of the input graph \(G\) can lead to exact polynomial algorithms and approximation algorithms. We also develop Fixed-Parameter Tractable \((\mathsf{FPT})\) algorithms for the problem parameterized by the “dual parameter” \(\ell_{\mathcal{C}}\), defined as the minimum number of vertices of \(G\) which are not kept in the solution.

MSC:

68R10 Graph theory (including graph drawing) in computer science
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q27 Parameterized complexity, tractability and kernelization
68W05 Nonnumerical algorithms
68W25 Approximation algorithms
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