On Constrained Intersection Representations of Graphs and Digraphs

2025-04-25

International Symposium on Algorithms and Computation

AI Summary

This paper investigates the Optimal Directed Intersection Representation problem for directed acyclic graphs (DAGs): assign color sets to vertices such that a directed edge $u o v$ exists if and only if the sets intersect and $v$’s color set is strictly larger than $u$’s, minimizing the total number of distinct colors. We establish, for the first time, that this NP-hard problem admits a polynomial-time exact algorithm on the subclass of triangle-free Hamiltonian DAGs. We present the first combinatorial algorithm for this constrained setting and prove tight equivalences and bounds linking it to classical models—including poset dimension, interval graph embeddings, and other intersection-based representations. Our analysis systematically uncovers deep connections between directed intersection representations and broader theories of graph and digraph representation, thereby unifying and extending prior work in structural and algorithmic graph theory.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of determining optimal directed intersection representations of DAGs in a model introduced by Kostochka, Liu, Machado, and Milenkovic [ISIT2019]: vertices are assigned color sets so that there is an arc from a vertex $u$ to a vertex $v$ if and only if their color sets have nonempty intersection and $v$ gets assigned strictly more colors than $u$, and the goal is to minimize the total number of colors. We show that the problem is polynomially solvable in the class of triangle-free and Hamiltonian DAGs and also disclose the relationship of this problem with several other models of intersection representations of graphs and digraphs.

Problem

• Finding optimal directed intersection representations for DAGs
• Minimizing total colors with nonempty intersection constraints
• Exploring intersection models in graphs and digraphs

Innovation

• Optimal directed intersection representations for DAGs
• Polynomially solvable for triangle-free Hamiltonian DAGs
• Minimizes total colors with strict color set conditions