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

TL;DR

The paper proves that finding a T-odd acyclic orientation on partially directed graphs is NP-complete, even when the underlying graph is planar with maximum degree $3$ and all vertices outside $T$ have degree $2$. The authors reduce Planar3-SAT to the problem by constructing variable and clause gadgets that enforce consistent boundary behavior and satisfiability constraints, establishing a tight correspondence between truth assignments and feasible orientations. This hardness result complements known polynomial-time cases (e.g., trees, certain planar/3-regular restrictions with $|Vackslash T|=1$) and Szegedy's randomized algorithm for the general problem, underscoring the challenge of acquiring simple greedy solutions. Overall, the work delineates a sharp boundary in the complexity of acyclic orientations with parity constraints and motivates further study of constrained orientation problems on special graph classes.

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 Király (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.

Figures (8)

• Figure 1: Incidence graph of a Planar3-SAT instance. Here, $\sigma_{c^4} = (x_2, x_5, x_4)$ and $\sigma_{x_5} = (c^5, c^3, c^4, c^2)$
• Figure 4: Construction of part of $I_{\mathcal{X}, \mathcal{C}}$ from a Planar3-SAT instance $(\mathcal{X}, \mathcal{C})$. Some of the gadgets are simplified.
• Figure 5: The two possible acyclic orientations of $\Gamma_M$
• Figure 6: $O(\Gamma_X)_{out}$
• Figure 7: $O(\Gamma_X)_{in}$

Theorems & Definitions (7)

• Proposition 1.1: 2.11.5 in szegedyApplicationsWeightedCombinatorial2005
• Theorem 1.2
• Theorem 3.1: Lichtenstein 1982 lichtensteinPlanarFormulaeTheir1982
• Lemma 3.2
• Corollary 3.3
• Lemma 3.4
• Theorem 3.5