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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity, each level y is equipped with a partial order ≺_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of ≺_y. Constrained Level Planarity is known to be a remarkably difficult problem: previous results by Klemz and Rote [ACM Trans. Alg.'19] and by Brückner and Rutter [SODA'17] imply that it remains NP-hard even when restricted to graphs whose tree-depth and feedback vertex set number are bounded by a constant and even when the instances are additionally required to be either proper, meaning that each edge spans two consecutive levels, or ordered, meaning that all given partial orders are total orders. In particular, these results rule out the existence of FPT-time (even XP-time) algorithms with respect to these and related graph parameters (unless P=NP). However, the parameterized complexity of Constrained Level Planarity with respect to the vertex cover number of the input graph remained open.
In this paper, we show that Constrained Level Planarity can be solved in FPT-time when parameterized by the vertex cover number. In view of the previous intractability statements, our result is best-possible in several regards: a speed-up to polynomial time or a generalization to the aforementioned smaller graph parameters is not possible, even if restricting to proper or ordered instances.

Boris Klemz and Marie Diana Sieper. Constrained Level Planarity Is FPT with Respect to the Vertex Cover Number. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 99:1-99:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{klemz_et_al:LIPIcs.ICALP.2024.99, author = {Klemz, Boris and Sieper, Marie Diana}, title = {{Constrained Level Planarity Is FPT with Respect to the Vertex Cover Number}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {99:1--99:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.99}, URN = {urn:nbn:de:0030-drops-202428}, doi = {10.4230/LIPIcs.ICALP.2024.99}, annote = {Keywords: Parameterized Complexity, Graph Drawing, Planar Poset Diagram, Level Planarity, Constrained Level Planarity, Vertex Cover, FPT, Computational Geometry} }

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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity (CLP), each level y is equipped with a partial order ≺_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of ≺_y. Ordered Level Planarity (OLP) corresponds to the special case of CLP where the given partial orders ≺_y are total orders. Previous results by Brückner and Rutter [SODA 2017] and Klemz and Rote [ACM Trans. Alg. 2019] state that both CLP and OLP are NP-hard even in severely restricted cases. In particular, they remain NP-hard even when restricted to instances whose width (the maximum number of vertices that may share a common level) is at most two. In this paper, we focus on the other dimension: we study the parameterized complexity of CLP and OLP with respect to the height (the number of levels).
We show that OLP parameterized by the height is complete with respect to the complexity class XNLP, which was first studied by Elberfeld, Stockhusen, and Tantau [Algorithmica 2015] (under a different name) and recently made more prominent by Bodlaender, Groenland, Nederlof, and Swennenhuis [FOCS 2021]. It contains all parameterized problems that can be solved nondeterministically in time f(k)⋅ n^O(1) and space f(k)⋅ log n (where f is a computable function, n is the input size, and k is the parameter). If a problem is XNLP-complete, it lies in XP, but is W[t]-hard for every t.
In contrast to the fact that OLP parameterized by the height lies in XP, it turns out that CLP is NP-hard even when restricted to instances of height 4. We complement this result by showing that CLP can be solved in polynomial time for instances of height at most 3.

Václav Blažej, Boris Klemz, Felix Klesen, Marie Diana Sieper, Alexander Wolff, and Johannes Zink. Constrained and Ordered Level Planarity Parameterized by the Number of Levels. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 20:1-20:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blazej_et_al:LIPIcs.SoCG.2024.20, author = {Bla\v{z}ej, V\'{a}clav and Klemz, Boris and Klesen, Felix and Sieper, Marie Diana and Wolff, Alexander and Zink, Johannes}, title = {{Constrained and Ordered Level Planarity Parameterized by the Number of Levels}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {20:1--20:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.20}, URN = {urn:nbn:de:0030-drops-199652}, doi = {10.4230/LIPIcs.SoCG.2024.20}, annote = {Keywords: Parameterized Complexity, Graph Drawing, XNLP, XP, W\lbrackt\rbrack-hard, Level Planarity, Planar Poset Diagram, Computational Geometry} }