On the Maximum Colorful Arborescence Problem and Color Hierarchy Graph Structure

Let G = (V, A) be a vertex-colored arc-weighted directed acyclic graph (DAG) rooted in some vertex r. The color hierarchy graph H(G) of G is defined as follows: V (H(G)) is the color set C of G, and H(G) has an arc from c to c if G has an arc from a vertex of color c to a vertex of color c. We study the Maximum Colorful Arborescence (MCA) problem, which takes as input a DAG G such that H(G) is also a DAG, and aims at finding in G a maximum-weight arborescence rooted in r in which no color appears more than once. The MCA problem models the de novo inference of unknown metabolites by mass spectrometry experiments. Although the problem has been introduced ten years ago (under a different name), it was only recently pointed out that a crucial additional property in the problem definition was missing: by essence, H(G) must be a DAG. In this paper, we further investigate MCA under this new light and provide new algorithmic results for this problem, with a specific focus on fixed-parameter tractability (FPT) issues for different structural parameters of H(G). In particular, we show there exists an O(3 ∗ H) time algorithm for solving MCA, where nH is the number of vertices of indegree at least two in H(G), thereby improving the O(3) algorithm from Böcker et al. [Proc. ECCB ’08]. We also prove that MCA is W[2]-hard relatively to the treewidth Ht of the underlying undirected graph of H(G), and further show that it is FPT relatively to Ht + lC , where lC := |V | − |C|. 2012 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory