eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-12-13
19:1
19:14
10.4230/LIPIcs.IPEC.2023.19
article
Finding Degree-Constrained Acyclic Orientations
Garvardt, Jaroslav
1
2
https://orcid.org/0000-0002-8762-8567
Renken, Malte
3
https://orcid.org/0000-0002-1450-1901
Schestag, Jannik
1
4
2
https://orcid.org/0000-0001-7767-2970
Weller, Mathias
3
https://orcid.org/0000-0002-9653-3690
Philipps-Universität Marburg, Germany
Friedrich-Schiller-Universität Jena, Germany
Technische Universität Berlin, Germany
Technische Universiteit Delft, The Netherlands
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].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol285-ipec2023/LIPIcs.IPEC.2023.19/LIPIcs.IPEC.2023.19.pdf
Graph Orientation
Phylogenetic Networks
General Factor
NP-hardness
Parameterized Algorithms
Treewidth