The Maximum Colorful Arborescence problem parameterized by the structure of its color hierarchy graph

TL;DR

This paper investigates the Maximum Colorful Arborescence problem in DAGs with a focus on the structure of the color hierarchy graph, providing new algorithms and complexity results based on structural parameters.

Contribution

The paper introduces an improved fixed-parameter algorithm for MCA based on the number of high indegree vertices in the color hierarchy graph and analyzes its complexity relative to graph parameters.

Findings

Developed an O*(3^{nhs}) algorithm for MCA.

Proved MCA is W[2]-hard with respect to the treewidth of H.

Showed MCA is fixed-parameter tractable relative to Ht+lc.

Abstract

Let G=(V,A) be a vertex-colored arc-weighted directed acyclic graph (DAG) rooted in some vertex r, and let H be its color hierarchy graph, defined as follows: V(H) is the color set C of G, and an arc from color c to color c' exists in H if there is an arc in G from a vertex of color c to a vertex of color c'. In this paper, we study the MAXIMUM COLORFUL ARBORESCENCE problem (or MCA), which takes as input a DAG G with the additional constraint that H is also a DAG, and aims at finding in G an arborescence rooted in r, of maximum weight, and in which no color appears more than once. The MCA problem is motivated by the inference of unknown metabolites from mass spectrometry experiments. However, whereas the problem has been studied for roughly ten years, the crucial property that H is necessarily a DAG has only been pointed out and exploited very recently. In this paper, we further investigate MCA under this new light, by providing algorithmic results for the problem, with a specific focus on fixed-parameterized tractability (FPT) issues, and relatively to different structural parameters of H. In particular, we provide an O*(3^{nhs}) time algorithm for solving MCA, where nhs is the number of vertices of indegree at least two in H, thereby improving the O*(3^{|C|}) algorithm from [B\"ocker et al. 2008]. We also prove that MCA is W[2]-hard relatively to the treewidth Ht of H, and further show that it is FPT relatively to Ht+lc, where lc = |V| - |C|.