Color-Constrained Arborescences in Edge-Colored Digraphs

Consider a multigraph $G$ whose edges are colored from $[q]$ ($q$-colored graph) and $\alpha=(\alpha_1,\ldots,\alpha_{q}) \in \mathbb{N}^{q}$ (color-constraint). A subgraph $H$ of $G$ is called $\alpha$-colored if $H$ has exactly $\alpha_i$ edges of color $i$ for each $i \in[q]$. In this paper, we focus on $\alpha$-colored arborescences (spanning out-trees) in $q$-colored multidigraphs. We study the decision, counting and search versions of this problem. It is known that the decision and search problems are polynomial-time solvable when $q=2$ [Barahona and Pulleyblank, Discret. Appl. Math. 1987] and that the decision problem is NP-complete when $q$ is arbitrary [Ardra et al., arXiv 2024]. However the complexity status of the problem for fixed $q$ was open for $q > 2$. We solve this problem using an algebraic approach. Given a $q$-colored digraph $G$ and a vertex $s$ in $G$, we construct a symbolic matrix in $q-1$ indeterminates such that the number of $\alpha$-colored arborescences in $G$ rooted at $s$ for all color-constraints $\alpha \in \mathbb{N}^q$ can be read from its determinant polynomial. This result extends Tutte's matrix-tree theorem and gives a polynomial-time algorithm for the counting and decision problems for fixed $q$. We use it to design an algorithm that finds an $\alpha$-colored arborescence when one exists. We also study the weighted variant of the problem and give a polynomial-time algorithm (when $q$ is fixed and weights are polynomially bounded) which finds a minimum weight solution.