Chakraborty, D. orcid.org/0000-0003-0534-6417, Dailly, A., Foucaud, F. et al. (1 more author) (2024) Algorithms and complexity for path covers of temporal DAGs. In: 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). 49th International Symposium on Mathematical Foundations of Computer Science, 26-30 Aug 2024, Bratislava, Slovakia. Leibniz International Proceedings in Informatics (LIPIcs), 306 . Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik , Wadern, Germany , 38:1-38:17. ISBN 978-3-95977-335-5
Abstract
A path cover of a digraph is a collection of paths collectively containing its vertex set. A path cover with minimum cardinality for a directed acyclic graph can be found in polynomial time [Fulkerson, AMS’56; Cáceres et al., SODA’22]. Moreover, Dilworth’s celebrated theorem on chain coverings of partially ordered sets equivalently states that the minimum size of a path cover of a DAG is equal to the maximum size of a set of mutually unreachable vertices. In this paper, we examine how far these classic results can be extended to a dynamic setting.
A temporal digraph has an arc set that changes over discrete time-steps; if the underlying digraph is acyclic, then it is a temporal DAG. A temporal path is a directed path in the underlying digraph, such that the time-steps of arcs are strictly increasing along the path. Two temporal paths are temporally disjoint if they do not occupy any vertex at the same time. A temporal path cover is a collection C of temporal paths that covers all vertices, and C is temporally disjoint if all its temporal paths are pairwise temporally disjoint. We study the computational complexities of the problems of finding a minimum-size temporal (disjoint) path cover (denoted as Temporal Path Cover and Temporally Disjoint Path Cover).
On the negative side, we show that both Temporal Path Cover and Temporally Disjoint Path Cover are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, Temporally Disjoint Path Cover remains NP-hard even on temporal oriented trees. We also observe that natural temporal analogues of Dilworth’s theorem on these classes of temporal DAGs do not hold.
In contrast, we show that Temporal Path Cover is polynomial-time solvable on temporal oriented trees by a reduction to Clique Cover for (static undirected) weakly chordal graphs (a subclass of perfect graphs for which Clique Cover admits an efficient algorithm). This highlights an interesting algorithmic difference between the two problems. Although it is NP-hard on temporal oriented trees, Temporally Disjoint Path Cover becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees.
Motivated by the hardness result on trees, we show that, in contrast, Temporal Path Cover admits an XP time algorithm with respect to parameter tmax + tw, where tmax is the maximum time-step and tw is the treewidth of the underlying static undirected graph; moreover, Temporally Disjoint Path Cover admits an FPT algorithm with respect to the same parameterization.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Dibyayan Chakraborty, Antoine Dailly, Florent Foucaud, and Ralf Klasing; licensed under Creative Commons License CC-BY 4.0 . |
Keywords: | Temporal Graphs, Dilworth’s Theorem, DAGs, Path Cover, Temporally Disjoint Paths, Algorithms, Oriented Trees, Treewidth |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) > Algorithms & Complexity |
Depositing User: | Symplectic Publications |
Date Deposited: | 22 Jul 2024 10:21 |
Last Modified: | 30 Sep 2024 09:57 |
Published Version: | https://drops.dagstuhl.de/entities/document/10.423... |
Status: | Published |
Publisher: | Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik |
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) |
Identification Number: | 10.4230/LIPIcs.MFCS.2024.38 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:214992 |
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