Note about the complexity of the acyclic orientation with parity constraint problem
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| Format: | Preprint |
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2025
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| _version_ | 1866908671886229504 |
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| author | Gravier, Sylvain Petiteau, Matthieu Sivignon, Isabelle |
| author_facet | Gravier, Sylvain Petiteau, Matthieu Sivignon, Isabelle |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_20935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note about the complexity of the acyclic orientation with parity constraint problem Gravier, Sylvain Petiteau, Matthieu Sivignon, Isabelle Discrete Mathematics Combinatorics 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. |
| title | Note about the complexity of the acyclic orientation with parity constraint problem |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2504.20935 |