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Documents authored by Zink, Johannes

Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2024.42
From Planar via Outerplanar to Outerpath – Engineering NP-Hardness Constructions (Poster Abstract)

Authors: Joshua Geis and Johannes Zink

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
A typical question in graph drawing is to determine, for a given graph drawing style, the boundary between polynomial-time solvability and NP-hardness. For two examples from the area of drawing graphs with few slopes, we sharpen this boundary. We suggest a framework for a certain type of NP-hardness constructions where graphs have some parts that can only be realized as rigid structures, whereas other parts allow a controllable degree of flexibility. Starting with an NP-complete problem involving planarity (here, we use planar monotone rectilinear 3-SAT), we consider first a reduction to a planar graph, which can be adjusted to an outerplanar graph, and finally to an outerpath. An outerplanar graph is a graph admitting an outerplanar drawing, that is, a crossing-free drawing where every vertex lies on the outer face, and an outerpath is a graph admitting an outerplanar drawing where the weak dual is a path. The (weak) dual of a graph drawing is the graph that has a node for every (inner) face and a link if two faces share an edge. Specifically, we first show that, for every upward-planar directed outerpath G, it is NP-hard to decide whether G admits an upward-planar straight-line drawing where every edge has one of three distinct slopes, and we second show that, for every undirected outerpath G, it is NP-hard to decide whether G admits a proper level-planar straight-line drawing where every edge has one of two distinct slopes. For both problems, NP-hardness has been known before for outerplanar graphs.
Cite as

Joshua Geis and Johannes Zink. From Planar via Outerplanar to Outerpath – Engineering NP-Hardness Constructions (Poster Abstract). In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 42:1-42:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{geis_et_al:LIPIcs.GD.2024.42,
  author =	{Geis, Joshua and Zink, Johannes},
  title =	{{From Planar via Outerplanar to Outerpath – Engineering NP-Hardness Constructions}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{42:1--42:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.42},
  URN =		{urn:nbn:de:0030-drops-213263},
  doi =		{10.4230/LIPIcs.GD.2024.42},
  annote =	{Keywords: NP-hardness, outerplanar, outerpath}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.20
Constrained and Ordered Level Planarity Parameterized by the Number of Levels

Authors: Václav Blažej, Boris Klemz, Felix Klesen, Marie Diana Sieper, Alexander Wolff, and Johannes Zink

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

Abstract
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.
Cite as

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}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.36
Coloring and Recognizing Mixed Interval Graphs

Authors: Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Felix Klesen, Paweł Rzążewski, Alexander Wolff, and Johannes Zink

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
A mixed interval graph is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph G, an interval u receives a lower (different) color than an interval v if G contains arc (u,v) (edge {u,v}). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a min {ω(G), λ(G)+1}-approximation algorithm, where ω(G) is the size of a largest clique and λ(G) is the length of a longest directed path in G. For the subclass of bidirectional interval graphs (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call containment interval graphs. In such a graph, there is an arc (u,v) if interval u contains interval v, and there is an edge {u,v} if u and v overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.
Cite as

Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Felix Klesen, Paweł Rzążewski, Alexander Wolff, and Johannes Zink. Coloring and Recognizing Mixed Interval Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gutowski_et_al:LIPIcs.ISAAC.2023.36,
  author =	{Gutowski, Grzegorz and Junosza-Szaniawski, Konstanty and Klesen, Felix and Rz\k{a}\.{z}ewski, Pawe{\l} and Wolff, Alexander and Zink, Johannes},
  title =	{{Coloring and Recognizing Mixed Interval Graphs}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{36:1--36:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.36},
  URN =		{urn:nbn:de:0030-drops-193388},
  doi =		{10.4230/LIPIcs.ISAAC.2023.36},
  annote =	{Keywords: Interval Graphs, Mixed Graphs, Graph Coloring}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.18
Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines

Authors: Martin Koutecký and Johannes Zink

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (Q|HM|C_max) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model (R|HM|C_max), we show that if either the largest job size p_max, or the number of jobs n are polynomially bounded in the instance size |I|, there are algorithms with complexity |I|^poly(k). Our main result is that this is unlikely to be improved, because Q||C_max is W[1]-hard parameterized by k already when n, p_max, and the numbers describing the speeds are polynomial in |I|; the same holds for R|HM|C_max (without speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives 𝓁₂-norm minimization of the load vector and, partially, sum of weighted completion times ∑ w_j C_j. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (P||C_max) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for Q||C_max this implies that the complexity of P|HM|C_max is the only remaining open case.
Cite as

Martin Koutecký and Johannes Zink. Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{koutecky_et_al:LIPIcs.ISAAC.2020.18,
  author =	{Kouteck\'{y}, Martin and Zink, Johannes},
  title =	{{Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.18},
  URN =		{urn:nbn:de:0030-drops-133620},
  doi =		{10.4230/LIPIcs.ISAAC.2020.18},
  annote =	{Keywords: Scheduling, cutting stock, hardness, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2020.35
Simplification of Polyline Bundles

Authors: Joachim Spoerhase, Sabine Storandt, and Johannes Zink

Published in: LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Abstract
We propose and study a generalization to the well-known problem of polyline simplification. Instead of a single polyline, we are given a set of l polylines possibly sharing some line segments and bend points. Our goal is to minimize the number of bend points in the simplified bundle with respect to some error tolerance δ (measuring Fréchet distance) but under the additional constraint that shared parts have to be simplified consistently. We show that polyline bundle simplification is NP-hard to approximate within a factor n^(1/3 - ε) for any ε > 0 where n is the number of bend points in the polyline bundle. This inapproximability even applies to instances with only l=2 polylines. However, we identify the sensitivity of the solution to the choice of δ as a reason for this strong inapproximability. In particular, we prove that if we allow δ to be exceeded by a factor of 2 in our solution, we can find a simplified polyline bundle with no more than 𝒪(log (l + n)) ⋅ OPT bend points in polytime, providing us with an efficient bi-criteria approximation. As a further result, we show fixed-parameter tractability in the number of shared bend points.
Cite as

Joachim Spoerhase, Sabine Storandt, and Johannes Zink. Simplification of Polyline Bundles. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{spoerhase_et_al:LIPIcs.SWAT.2020.35,
  author =	{Spoerhase, Joachim and Storandt, Sabine and Zink, Johannes},
  title =	{{Simplification of Polyline Bundles}},
  booktitle =	{17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-150-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{162},
  editor =	{Albers, Susanne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.35},
  URN =		{urn:nbn:de:0030-drops-122821},
  doi =		{10.4230/LIPIcs.SWAT.2020.35},
  annote =	{Keywords: Polyline Simplification, Bi-criteria Approximation, Hardness of Approximation, Geometric Set Cover}
}
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