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 [1] 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.

给定一个顶点着色的弧加权有向无环图G，最大彩色子树问题（MCS）的目的是在G中找到最大权重的树状结构，其中颜色不止出现一次。该问题最初是在[1]中通过串联质谱从头鉴定代谢物的背景下引入的。但是，对初始动机的透彻分析表明，应该修改MCS的正式定义，因为输入图G实际上具有额外的属性，这些属性到目前为止尚未开发。这使我们在本文中描述了一个更精确的模型，称为最大彩色树状结构（MCA），我们在算法复杂度方面进行了广泛的研究。特别地，我们表明，利用输入图G的隐式颜色层次图可以导致精确的多项式算法和逼近算法。我们还开发了固定参数可牵引式（$\mathsf{FPT}$）针对由“双重参数”参数化的问题的算法 ${\ell }_{\mathcal{C}}$，其定义为顶点的最小数目ģ其不保持在溶液中。