Color-Constrained Arborescences in Edge-Colored Digraphs

TL;DR

This work studies color-constrained arborescences in $q$-colored digraphs by introducing color-sensitive symbolic Laplacians $L_G$ and $L'_G$ with $q-1$ indeterminates (and $x_q=1$). The main result shows that the coefficient of $x^{\alpha}$ in the determinant of the appropriate reduced Laplacian counts $\alpha$-colored $s$-arborescences, extending Tutte's matrix-tree theorem to the colored setting and enabling counting for all $\alpha$ in one shot for fixed $q$. Consequently, the authors provide polynomial-time algorithms for counting, decision, and finding an $\alpha$-colored $s$-arborescence when $q$ is fixed, and extend the framework to the weighted case to obtain minimum-weight color-constrained arborescences using determinant-based techniques and number-theoretic implementation details. The paper also discusses practical aspects such as Evaluate-Interpolate determinant computation and modular arithmetic, and highlights open questions on fixed-parameter tractability in $q$ and enumeration, bridging algebraic matrix-tree methods with matroid intersection problems.

Abstract

Given a multigraph $G$ whose edges are colored from the set $[q]:=\{1,2,\ldots,q\}$ (\emph{$q$-colored graph}), and a vector $α=(α_1,\ldots,α_{q}) \in \mathbb{N}^{q}$ (\emph{color-constraint}), a subgraph $H$ of $G$ is called \emph{$α$-colored}, if $H$ has exactly $α_i$ edges of color $i$ for each $i \in[q]$. In this paper, we focus on $α$-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 $q=2$ and that the decision problem is NP-complete when $q$ is arbitrary. However the complexity status of the problem for fixed $q$ was open for $q > 2$. We show that, for a $q$-colored digraph $G$ and a vertex $s$ in $G$, the number of $α$-colored arborescences in $G$ rooted at $s$ for all color-constraints $α\in \mathbb{N}^q$ can be read from the determinant of a symbolic matrix in $q-1$ indeterminates. This result extends Tutte's matrix-tree theorem for directed graphs and gives a polynomial-time algorithm for the counting and decision problems for fixed $q$. We also use it to design an algorithm that finds an $α$-colored arborescence when one exists. Finally, we study the weighted variant of the problem and give a polynomial-time algorithm (when $q$ is fixed) which finds a minimum weight solution.

Theorems & Definitions (17)

• Definition 1
• Theorem 2: Extended Tutte's Theorem
• Theorem 3: Corollary 4.1 in Moon94
• Lemma 4
• proof
• Lemma 5
• Definition 6
• Theorem 7: Extended Kirchhoff's Theorem
• Theorem 8
• Lemma 9