Finding Degree-Constrained Acyclic Orientations

Authors Jaroslav Garvardt , Malte Renken , Jannik Schestag , Mathias Weller

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Author Details

Jaroslav Garvardt
  • Philipps-Universität Marburg, Germany
  • Friedrich-Schiller-Universität Jena, Germany
Malte Renken
  • Technische Universität Berlin, Germany
Jannik Schestag
  • Philipps-Universität Marburg, Germany
  • Technische Universiteit Delft, The Netherlands
  • Friedrich-Schiller-Universität Jena, Germany
Mathias Weller
  • Technische Universität Berlin, Germany


This work was initiated on the research retreat of the Algorithmics and Computational Complexity group of TU Berlin, held in Darlingerode in September 2022. We thank André Nichterlein and Till Fluschnik for helpful discussions and ideas and our anonymous reviewers for pointing out many small errors in previous versions of the paper.

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Jaroslav Garvardt, Malte Renken, Jannik Schestag, and Mathias Weller. Finding Degree-Constrained Acyclic Orientations. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We consider the problem of orienting a given, undirected graph into a (directed) acyclic graph such that the in-degree of each vertex v is in a prescribed list λ(v). Variants of this problem have been studied for a long time and with various applications, but mostly without the requirement for acyclicity. Without this requirement, the problem is closely related to the classical General Factor problem, which is known to be NP-hard in general, but polynomial-time solvable if no list λ(v) contains large "gaps" [Cornuéjols, J. Comb. Theory B, 1988]. In contrast, we show that deciding if an acyclic orientation exists is NP-hard even in the absence of such "gaps". On the positive side, we design parameterized algorithms for various, natural parameterizations of the acyclic orientation problem. A special case of the orientation problem with degree constraints recently came up in the context of reconstructing evolutionary histories (that is, phylogenetic networks). This phylogenetic setting imposes additional structure onto the problem that can be exploited algorithmically, allowing us to show fixed-parameter tractability when parameterized by either the treewidth of G (a smaller parameter than the frequently employed "level"), by the number of vertices v for which |λ(v)| ≥ 2, by the number of vertices v for which the highest value in λ(v) is at least 2. While the latter result can be extended to the general degree-constraint acyclic orientation problem, we show that the former cannot unless FPT=W[1].

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Graph Orientation
  • Phylogenetic Networks
  • General Factor
  • NP-hardness
  • Parameterized Algorithms
  • Treewidth


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  1. Ashwin Arulselvan, Ágnes Cseh, Martin Groß, David F. Manlove, and Jannik Matuschke. Matchings with lower Quotas: Algorithms and Complexity. Algorithmica, 80(1):185-208, 2018. URL:
  2. Laurent Bulteau and Mathias Weller. Parameterized Algorithms in Bioinformatics: An Overview. Algorithms, 12(12):256, 2019. URL:
  3. Laurent Bulteau, Mathias Weller, and Louxin Zhang. On turning a Graph into a Phylogenetic Network. working paper or preprint, 2019. URL:
  4. Gérard Cornuéjols. General Factors of Graphs. Journal of Combinatorial Theory Series B, 45(2):185-198, 1988. URL:
  5. Joseph Felsenstein. Inferring Phylogenies. Sinauer Associates, 2 edition, 2003. Google Scholar
  6. András Frank. On the Orientation of Graphs. Journal of Combinatorial Theory Series B, 28(3):251-261, 1980. URL:
  7. Harold N. Gabow. Upper degree-constrained partial Orientations. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 554-563, 2006. URL:
  8. András Gyárfás and András Frank. How to orient the edges of a graph. In Proceedings of the Fifth Hungarian Combinatorial Colloquium, Keszthely, pages 353-362, 1976. URL:
  9. Katharina T. Huber, Leo van Iersel, Remie Janssen, Mark Jones, Vincent Moulton, Yukihiro Murakami, and Charles Semple. Orienting undirected Phylogenetic Networks, 2019. URL:
  10. Daniel H. Huson, Regula Rupp, and Céline Scornavacca. Phylogenetic Networks - Concepts, Algorithms and Applications. Cambridge University Press, 2010. Google Scholar
  11. Zoltán Király and Dömötör Pálvölgyi. Acyclic orientations with degree constraints, 2018. URL:
  12. László Lovász. The Factorization of Graphs. In Proceedings of the Calgary International Conference on Combinatorial Structures and their Applications, 1970. Google Scholar
  13. László Lovász. The Factorization of Graphs. II. Acta Mathematica Academiae Scientiarum Hungarica, 23:223-246, 1972. URL:
  14. Luke Mathieson and Stefan Szeider. Editing Graphs to satisfy degree constraints: A parameterized approach. Journal of Computer and System Sciences, 78(1):179-191, 2012. URL:
  15. Erin K Molloy, Arun Durvasula, and Sriram Sankararaman. Advancing admixture Graph estimation via maximum likelihood Network Orientation. Bioinformatics, 37(Supplement 1):i142-i150, July 2021. URL:
  16. Thomas J. Schaefer. The Complexity of Satisfiability Problems. In Proceedings of the 10th Annual ACM symposium on Theory of computing (STOC), pages 216-226, 1978. URL:
  17. Feng Shi, Qilong Feng, Jianer Chen, Lusheng Wang, and Jianxin Wang. Distances between Phylogenetic Trees: A survey. Tsinghua Science and Technology, 18(5):490-499, 2013. URL:
  18. Katherine St. John. Review Paper: The shape of Phylogenetic Treespace. Systematic Biology, 66(1):e83-e94, June 2016. URL: