Some polynomial classes for the acyclic orientation with parity constraint problem

2026-03-10

AI Summary

This study addresses the problem of finding an acyclic orientation of an undirected graph such that a specified vertex subset \( T \) has odd in-degree, a question whose computational complexity remained open after imposing acyclicity. By analyzing the global parity condition \( P \) together with source and sink existence conditions \( S \) and \( \bar{S} \), the authors establish for the first time a hierarchical classification of graph classes based on combinations of these conditions and rigorously prove their inclusion relationships. For several graph classes, they show that certain necessary conditions are also sufficient and develop polynomial-time algorithms—built upon the framework of Frank and Király—to construct feasible \( T \)-odd acyclic orientations. In particular, they provide a complete characterization of solvability for Cartesian products of paths and cycles, and demonstrate strict containments among the defined classes using cliques and product graphs, with all solvable instances admitting efficient constructions.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the problem of finding an acyclic orientation of an undirected graph with constrained in-degree parities specified by a subset T of vertices. An orientation is called T -odd if a vertex v has odd in-degree if and only if v P T . While the unconstrained parity orientation problem is polynomial (Chevalier et al. (1983)), imposing acyclicity makes it more challenging, and its complexity remains an open question. Szegedy and Szegedy ( 2006) proposed a randomized polynomial-time algorithm for this problem, but it is not known whether it belongs to co-NP. Furthermore, Gravier et al. (2025) showed the problem becomes NP-complete on partially directed graphs, even when restricted to planar cubic graphs. We identify three necessary conditions for the existence of acyclic T -odd orientation: a global parity condition P, and two conditions S and S ensuring the existence of potential sources and sinks. Following the work of Frank and Kiraly (2002), we define graph classes containing the graphs for which a given subset of the necessary conditions P, S and S is also sufficient for the existence of an acyclic T -odd orientation. We establish the inclusion relationships between these classes. We complete the study of these classes by a characterization of the solvable instances for Cartesian products of paths and cycles. The proofs of these results are all constructive, so that acyclic T -odd orientations can be built in polynomial time whenever they exist. We use these families, along with cliques, to demonstrate the strictness of the class inclusions in our hierarchy.

Problem

• acyclic orientation
• parity constraint
• in-degree parity
• graph orientation
• T-odd orientation

Innovation

• acyclic orientation
• parity constraint
• polynomial-time algorithm
• graph classes
• constructive characterization