The Maximum Colorful Arborescence problem: How (computationally) hard can it be?

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 which no color appears more than once. The was originally introduced [1] context de novo identification metabolites by tandem mass spectrometry. However, thorough analysis initial motivation shows that formal definition should be amended, since input actually possesses extra properties, have been unexploited so far. This leads us to describe this paper precise model we call Arborescence ( MCA ), extensively study terms algorithmic complexity. In particular, show exploiting implied Color Hierarchy Graph can lead exact polynomial algorithms and approximation algorithms. We also develop Fixed-Parameter Tractable FPT for parameterized “dual parameter” ? C defined as minimum number vertices are not kept solution.