Algorithmic aspects of the maximum colorful arborescence problem

Abstract

Given a vertex-colored arc-weighted directed acyclic graph G, the Maximum Colorful Subtree problem (or MCS) aims at find-ing an arborescence of maximum weight in G, in which no color appears more than once. The problem was originally introduced in [2] in the context of de novo identification of metabolites by tandem mass spec-trometry. 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 unex-ploited. This leads us to introduce in this paper a more precise model that we call Maximum Colorful Arborescence (MCA), and exten-sively 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”, defined as the number of vertices of G which are not kept in the solution.