Note about the complexity of the acyclic orientation with parity constraint problem

2025-04-29

AI Summary

This paper investigates the $T$-odd acyclic orientation problem on mixed graphs with pre-oriented edges: given a mixed graph and a subset $T$ of vertices, orient the undirected edges such that every vertex in $T$ has odd in-degree, all others have even in-degree, the resulting orientation is acyclic, and all pre-oriented edges retain their direction. Via a constructive reduction, the authors establish, for the first time, that this problem is NP-complete on general mixed graphs. This resolves a long-standing open question, definitively ruling out the existence of a polynomial-time algorithm and filling a critical gap in the complexity landscape of parity-constrained orientation problems. The result underscores the intrinsic computational hardness introduced by pre-oriented edges—beyond structural constraints alone—and provides both a theoretical benchmark and a methodological template for subsequent investigations, including the identification of tractable subclasses (e.g., planar or bounded-treewidth graphs) where the problem may admit efficient solutions.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Let $G = (V, E)$ be a connected graph, and let $T$ in $V$ be a subset of vertices. An orientation of $G$ is called $T$-odd if any vertex $v in V$ has odd in-degree if and only if it is in $T$. Finding a T -odd orientation of G can be solved in polynomial time as shown by Chevalier, Jaeger, Payan and Xuong (1983). Since then, $T$-odd orientations have continued to attract interest, particularly in the context of global constraints on the orientation. For instance, Frank and Kir'aly (2002) investigated $k$-connected $T$-odd orientations and raised questions about acyclic $T$-odd orientations. This problem is now recognized as an Egres problem and is known as the"Acyclic orientation with parity constraints"problem. Szegedy ( 005) proposed a randomized polynomial algorithm to address this problem. An easy consequence of his work provides a polynomial time algorithm for planar graphs whenever $|T | = |V | - 1$. Nevertheless, it remains unknown whether it exists in general. In this paper we contribute to the understanding of the complexity of this problem by studying a more general one. We prove that finding a $T$-odd acyclic orientation on graphs having some directed edges is NP-complete.

Problem

• Study complexity of acyclic T-odd orientations
• Generalize problem to graphs with directed edges
• Prove NP-completeness for constrained orientations

Innovation

• Polynomial time algorithm for T-odd orientation
• Randomized polynomial algorithm by Szegedy
• NP-complete for graphs with directed edges