Color-Constrained Arborescences in Edge-Colored Digraphs

2025-03-18

AI Summary

This paper investigates the decision, exact counting, and explicit construction of color-constrained root-directed spanning trees (α-color arborescences) in *q*-colored multidigraphs, where *q* ≥ 3 is a fixed constant. We generalize Tutte’s directed Matrix-Tree Theorem to the multicolor setting by constructing a (*q*−1)-variate symbolic matrix whose determinant precisely enumerates all α-color arborescences. Combining algebraic combinatorics, dynamic programming, and weighted bipartite matching techniques, we design a polynomial-time algorithm that uniformly solves the decision, exact counting, and constructive variants—and extends to the minimum-weight version. Our work fully characterizes the computational complexity boundary for this problem under fixed *q*, thereby resolving a long-standing open question for *q* > 2.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Given a multigraph $G$ whose edges are colored from the set $[q]:={1,2,ldots,q}$ (emph{$q$-colored graph}), and a vector $alpha=(alpha_1,ldots,alpha_{q}) in mathbb{N}^{q}$ (emph{color-constraint}), a subgraph $H$ of $G$ is called emph{$alpha$-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 $q=2$ 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 $alpha$-colored arborescences in $G$ rooted at $s$ for all color-constraints $alpha in mathbb{N}^q$ 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 $alpha$-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.

Problem

• Study α-colored arborescences in q-colored multidigraphs.
• Determine complexity for fixed q > 2 in decision and search problems.
• Develop polynomial-time algorithms for counting and weighted variants.

Innovation

• Extends Tutte's matrix-tree theorem for directed graphs
• Uses symbolic matrix determinant for counting arborescences
• Provides polynomial-time algorithms for fixed color constraints