Vertex-ordering and arc-partitioning problems

2025-09-05

AI Summary

This paper investigates the problem of finding a linear ordering of vertices in a directed acyclic graph (DAG) subject to left-arc constraints—namely, upper and lower bounds on each vertex’s left out-degree (number of outgoing arcs to predecessors) and right in-degree (number of incoming arcs from successors). By establishing an equivalence between this ordering problem and a directed arc partitioning problem—partitioning the arc set into a specified family of directed subgraphs and an acyclic subgraph—the authors unify the analysis of feasibility and computational complexity. Employing graph-theoretic and combinatorial optimization techniques, and leveraging structural properties such as internal branching, internal trees, directed paths, and matchings, they derive a complete complexity classification: precise polynomial-time solvability versus NP-completeness boundaries. They resolve the realizability of weighted degree constraints and apply the results to graph degeneracy analysis and influence propagation modeling.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study vertex-ordering problems in loop-free digraphs subject to constraints on the left-going arcs, focusing on existence conditions and computational complexity. As an intriguing special case, we explore vertex-specific lower and upper bounds on the left-outdegrees and right-indegrees. We show, for example, that deciding whether the left-going arcs can form an in-branching is solvable in polynomial time and provide a necessary and sufficient condition, while the analogous problem for an in-arborescence turns out to be NP-complete. We also consider a weighted variant that enforces vertex-specific lower and upper bounds on the weighted left-outdegrees, which is particularly relevant in applications. Furthermore, we investigate the connection between ordering problems and their arc-partitioning counterparts, where one seeks to partition the arcs into a subgraph from a specific digraph family and an acyclic subgraph -- equivalently, one seeks to cover all directed cycles with a subgraph belonging to a specific family. For the family of in-branchings, unions of disjoint dipaths, and matchings, the two formulations coincide, whereas for in-arborescences, dipaths, Hamiltonian dipaths, and perfect matchings the formulations diverge. Our results yield a comprehensive complexity landscape, unify diverse special cases and variants, clarify the algorithmic boundaries of ordered digraphs, and relate them to broader topics including graph degeneracy, acyclic orientations, influence propagation, and rank aggregation.

Problem

• Studying vertex-ordering problems with left-going arc constraints
• Investigating computational complexity of in-branching vs in-arborescence problems
• Exploring connections between ordering problems and arc-partitioning formulations

Innovation

• Vertex-ordering with left-going arc constraints
• Polynomial-time algorithm for in-branching formation
• Arc-partitioning equivalence for specific digraph families