DROPS

Document

DOI: 10.4230/LIPIcs.IPEC.2025.15

New Algorithm for Combinatorial n-Folds and Applications

Authors: Klaus Jansen, Kai Kahler, Lis Pirotton, and Malte Tutas

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract

Block-structured integer linear programs (ILPs) play an important role in various application fields. We address n-fold ILPs where the matrix 𝒜 has a specific structure, i.e., where the blocks in the lower part of 𝒜 consist only of the row vectors (1,… ,1). In this paper, we propose an approach tailored to exactly these combinatorial n-folds. We utilize a divide and conquer approach to separate the original problem such that the right-hand side iteratively decreases in size. We show that this decrease in size can be calculated such that we only need to consider a bounded amount of possible right-hand sides. This, in turn, lets us efficiently combine solutions of the smaller right-hand sides to solve the original problem. We can decide the feasibility of, and also optimally solve, such problems in time (n r Δ)^O(r) log(‖b‖_∞), where n is the number of blocks, r the number of rows in the upper blocks and Δ = ‖A‖_∞. We complement the algorithm by discussing applications of the n-fold ILPs with the specific structure we require. We consider the problems of (i) scheduling on uniform machines, (ii) closest string and (iii) (graph) imbalance. Regarding (i), our algorithm results in running times of p_max^O(d)|I|^O(1), matching a lower bound derived via ETH. For (ii) we achieve running times matching the current state-of-the-art in the general case. In contrast to the state-of-the-art, our result can leverage a bounded number of column-types to yield an improved running time. For (iii), we improve the parameter dependency on the size of the vertex cover.

Cite as

Klaus Jansen, Kai Kahler, Lis Pirotton, and Malte Tutas. New Algorithm for Combinatorial n-Folds and Applications. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jansen_et_al:LIPIcs.IPEC.2025.15,
  author =	{Jansen, Klaus and Kahler, Kai and Pirotton, Lis and Tutas, Malte},
  title =	{{New Algorithm for Combinatorial n-Folds and Applications}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{15:1--15:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.15},
  URN =		{urn:nbn:de:0030-drops-251472},
  doi =		{10.4230/LIPIcs.IPEC.2025.15},
  annote =	{Keywords: integer linear programming, n-fold, parameterized complexity, scheduling, uniform machines}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.16

A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth

Authors: Sergio Cabello, Alexander Dobler, Gašper Fijavž, Thekla Hamm, and Mirko H. Wagner

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets 𝒮 of crossing types gives a recognition problem: does a given graph admit an 𝒮-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in 𝒮? We show that there is a set 𝒮_bad with three crossing types and the following properties: - If 𝒮 contains no crossing type from 𝒮_bad, then the recognition of graphs that admit an 𝒮-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. - If 𝒮 contains any crossing type from 𝒮_bad, then it is NP-hard to decide whether a graph has an 𝒮-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth.

Cite as

Sergio Cabello, Alexander Dobler, Gašper Fijavž, Thekla Hamm, and Mirko H. Wagner. A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cabello_et_al:LIPIcs.ISAAC.2025.16,
  author =	{Cabello, Sergio and Dobler, Alexander and Fijav\v{z}, Ga\v{s}per and Hamm, Thekla and Wagner, Mirko H.},
  title =	{{A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.16},
  URN =		{urn:nbn:de:0030-drops-249248},
  doi =		{10.4230/LIPIcs.ISAAC.2025.16},
  annote =	{Keywords: 1-planar, crossing type, treewidth, pathwidth}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.23

The Price of Connectivity Augmentation on Planar Graphs

Authors: Hugo A. Akitaya, Justin Dallant, Erik D. Demaine, Michael Kaufmann, Linda Kleist, Frederick Stock, Csaba D. Tóth, and Torsten Ueckerdt

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

Given two classes of graphs, 𝒢₁ ⊆ 𝒢₂, and a c-connected graph G ∈ 𝒢₁, we wish to augment G with a smallest cardinality set of new edges F to obtain a k-connected graph G' = (V,E∪ F) ∈ 𝒢₂. In general, this is the c → k connectivity augmentation problem. Previous research considered variants where 𝒢₁ = 𝒢₂ is the class of planar graphs, plane graphs, or planar straight-line graphs. In all three settings, we prove that the c → k augmentation problem is NP-complete when 2 ≤ c < k ≤ 5. However, the connectivity of the augmented graph G' is at most 5 if 𝒢₂ is limited to planar graphs. We initiate the study of the c → k connectivity augmentation problem for arbitrary k ∈ ℕ, where 𝒢₁ is the class of planar graphs, plane graphs, or planar straight-line graphs, and 𝒢₂ is a beyond-planar class of graphs: 𝓁-planar, 𝓁-plane topological, or 𝓁-plane geometric graphs. We obtain tight bounds on the tradeoffs between the desired connectivity k and the local crossing number 𝓁 of the augmented graph G'. We also show that our hardness results apply to this setting. The connectivity augmentation problem for triangulations is intimately related to edge flips; and the minimum augmentation problem to the flip distance between triangulations. We prove that it is NP-complete to find the minimum flip distance between a given triangulation and a 4-connected triangulation, settling an open problem posed in 2014, and present an EPTAS for this problem.

Cite as

Hugo A. Akitaya, Justin Dallant, Erik D. Demaine, Michael Kaufmann, Linda Kleist, Frederick Stock, Csaba D. Tóth, and Torsten Ueckerdt. The Price of Connectivity Augmentation on Planar Graphs. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{a.akitaya_et_al:LIPIcs.GD.2025.23,
  author =	{A. Akitaya, Hugo and Dallant, Justin and Demaine, Erik D. and Kaufmann, Michael and Kleist, Linda and Stock, Frederick and T\'{o}th, Csaba D. and Ueckerdt, Torsten},
  title =	{{The Price of Connectivity Augmentation on Planar Graphs}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{23:1--23:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.23},
  URN =		{urn:nbn:de:0030-drops-250095},
  doi =		{10.4230/LIPIcs.GD.2025.23},
  annote =	{Keywords: connectivity augmentation, local crossing number, flip distance}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.18

Internally-Convex Drawings of Outerplanar Graphs in Small Area

Authors: Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Giuseppe Liotta, and Antonios Symvonis

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

A well-known result by Kant [Algorithmica, 1996] implies that n-vertex outerplane graphs admit embedding-preserving planar straight-line grid drawings where the internal faces are convex polygons in O(n²) area. In this paper, we present an algorithm to compute such drawings in O(n¹·⁵) area. We also consider outerplanar drawings in which the internal faces are required to be strictly-convex polygons. In this setting, we consider outerplanar graphs whose weak dual is a path and give a drawing algorithm that achieves Θ(nk²) area, where k is the maximum size of an internal facial cycle.

Cite as

Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Giuseppe Liotta, and Antonios Symvonis. Internally-Convex Drawings of Outerplanar Graphs in Small Area. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bekos_et_al:LIPIcs.GD.2025.18,
  author =	{Bekos, Michael A. and Da Lozzo, Giordano and Frati, Fabrizio and Liotta, Giuseppe and Symvonis, Antonios},
  title =	{{Internally-Convex Drawings of Outerplanar Graphs in Small Area}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.18},
  URN =		{urn:nbn:de:0030-drops-250042},
  doi =		{10.4230/LIPIcs.GD.2025.18},
  annote =	{Keywords: Grid drawings, convexity, area bounds, outerplanar graphs}
}

Document

Poster Abstract

DOI: 10.4230/LIPIcs.GD.2025.48

Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid (Poster Abstract)

Authors: Henry Förster, Stephen Kobourov, Jacob Miller, and Johannes Zink

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

A strengthened version of Harborth’s well-known conjecture - known as Kleber’s conjecture - states that every planar graph admits a planar straight-line drawing where every edge has integer length and each vertex is restricted to the integer grid. Positive results for Kleber’s conjecture are known for planar 3-regular graphs, for planar graphs that have maximum degree 4, and for planar 3-trees. However, all but one of the existing results are existential and do not provide bounds on the required grid size. We provide polynomial-time algorithms for computing crossing-free straight-line drawings of trees and cactus graphs with integer edge lengths and integer vertex position on polynomial-size integer grids. We also give an historic overview of planar straight-line graph drawing results.

Cite as

Henry Förster, Stephen Kobourov, Jacob Miller, and Johannes Zink. Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 48:1-48:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{forster_et_al:LIPIcs.GD.2025.48,
  author =	{F\"{o}rster, Henry and Kobourov, Stephen and Miller, Jacob and Zink, Johannes},
  title =	{{Drawing Trees and Cacti with Integer Edge Lengths on a Polynomial-Size Grid}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{48:1--48:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.48},
  URN =		{urn:nbn:de:0030-drops-250349},
  doi =		{10.4230/LIPIcs.GD.2025.48},
  annote =	{Keywords: Harborth’s conjecture, tree drawings, cactus drawings, grid drawings}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.14

OOPS: Optimized One-Planarity Solver via SAT

Authors: Sergey Pupyrev

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

We present OOPS (Optimized One-Planarity Solver), a practical heuristic for recognizing 1-planar graphs and several important subclasses. A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once - a natural generalization of planar graphs that has received increasing attention in graph drawing and beyond-planar graph theory. Although testing planarity can be done in linear time, recognizing 1-planar graphs is NP-complete, making effective practical algorithms especially valuable. The core idea of our approach is to reduce the recognition of 1-planarity to a propositional satisfiability (SAT) instance, enabling the use of modern SAT solvers to efficiently explore the search space. Despite the inherent complexity of the problem, our method is substantially faster in practice than naïve or brute-force algorithms. In addition to demonstrating the empirical performance of our solver on synthetic and real-world instances, we show how OOPS can be used as a discovery tool in theoretical graph theory. Specifically, we employ OOPS to investigate two research problems concerning 1-planarity of specific graph families. Our implementation of the algorithm is publicly available to support further exploration in the field.

Cite as

Sergey Pupyrev. OOPS: Optimized One-Planarity Solver via SAT. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{pupyrev:LIPIcs.GD.2025.14,
  author =	{Pupyrev, Sergey},
  title =	{{OOPS: Optimized One-Planarity Solver via SAT}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.14},
  URN =		{urn:nbn:de:0030-drops-250004},
  doi =		{10.4230/LIPIcs.GD.2025.14},
  annote =	{Keywords: beyond planarity, 1-planar graph, SAT, book embeddings, upward 1-planarity}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.5

Constrained Flips in Plane Spanning Trees

Authors: Oswin Aichholzer, Joseph Dorfer, and Birgit Vogtenhuber

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

A flip in a plane spanning tree T is the operation of removing one edge from T and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1) Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of 2n-O(√n) on the diameter of the compatible flip graph to (5n/3)-O(1), by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA 2025] up to an additive constant of 1. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2) Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of 2n-O(1) for the diameter of the rotation graph to (7n/4)-O(1).

Cite as

Oswin Aichholzer, Joseph Dorfer, and Birgit Vogtenhuber. Constrained Flips in Plane Spanning Trees. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aichholzer_et_al:LIPIcs.GD.2025.5,
  author =	{Aichholzer, Oswin and Dorfer, Joseph and Vogtenhuber, Birgit},
  title =	{{Constrained Flips in Plane Spanning Trees}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.5},
  URN =		{urn:nbn:de:0030-drops-249913},
  doi =		{10.4230/LIPIcs.GD.2025.5},
  annote =	{Keywords: Non-crossing spanning trees, Flip Graphs, Diameter, Complexity, Happy edges}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.40

An Algorithm for Accurate and Simple-Looking Metaphorical Maps

Authors: Eleni Katsanou, Tamara Mchedlidze, Antonios Symvonis, and Thanos Tolias

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

Metaphorical maps or contact representations are visual representations of vertex-weighted graphs that rely on the geographic map metaphor. The vertices are represented by countries, the weights by the areas of the countries, and the edges by contacts/boundaries among them. The accuracy with which the weights are mapped to areas and the simplicity of the polygons representing the countries are the two classical optimization goals for metaphorical maps. Mchedlidze & Schnorr [Mchedlidze and Schnorr, 2022] presented a force-based algorithm that creates metaphorical maps that balance between these two optimization goals. Their maps look visually simple, but the accuracy of the maps is far from optimal - the countries' areas can vary up to 30% compared to required. In this paper, we provide a multi-fold extension of the algorithm in [Mchedlidze and Schnorr, 2022]. More specifically: 1) Towards improving accuracy: We introduce the notion of region stiffness and suggest a technique for varying the stiffness based on the current pressure of map regions. 2) Towards maintaining simplicity: We introduce a weight coefficient to the pressure force exerted on each polygon point based on whether the corresponding point appears along a narrow passage. 3) Towards generality: We cover, in contrast to [Mchedlidze and Schnorr, 2022], non-triangulated graphs. This is done by either generating points where more than three regions meet or by introducing holes in the metaphorical map. We perform an extended experimental evaluation that, among other results, reveals that our algorithm is able to construct metaphorical maps with nearly perfect area accuracy with a little sacrifice in their simplicity.

Cite as

Eleni Katsanou, Tamara Mchedlidze, Antonios Symvonis, and Thanos Tolias. An Algorithm for Accurate and Simple-Looking Metaphorical Maps. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{katsanou_et_al:LIPIcs.GD.2025.40,
  author =	{Katsanou, Eleni and Mchedlidze, Tamara and Symvonis, Antonios and Tolias, Thanos},
  title =	{{An Algorithm for Accurate and Simple-Looking Metaphorical Maps}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{40:1--40:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.40},
  URN =		{urn:nbn:de:0030-drops-250268},
  doi =		{10.4230/LIPIcs.GD.2025.40},
  annote =	{Keywords: Metaphorical maps, contact representation, accuracy (cartographic error), simplicity (polygon complexity), force directed algorithm}
}

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Poster Abstract

DOI: 10.4230/LIPIcs.GD.2025.47

Reconfigurations of Plane Caterpillars and Paths (Poster Abstract)

Authors: Todor Antić, Guillermo Gamboa Quintero, and Jelena Glišić

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

Let S be a point set in the plane, and let 𝒫(S) and 𝒞(S) be the sets of all plane spanning paths and caterpillars on S. We study reconfiguration operations on 𝒫(S) and 𝒞(S). In particular, we prove that all of the commonly studied reconfigurations on plane spanning trees still yield connected reconfiguration graphs for caterpillars when S is in convex position. If S is in general position, we show that the rotation, compatible flip and flip graphs of 𝒞(S) are connected while the slide graph is sometimes disconnected, but always has a component of size 1/4(3ⁿ-1). We then study sizes of connected components in reconfiguration graphs of plane spanning paths. In this direction, we show that no component of size at most 7 can exist in the flip graph on 𝒫(S).

Cite as

Todor Antić, Guillermo Gamboa Quintero, and Jelena Glišić. Reconfigurations of Plane Caterpillars and Paths (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 47:1-47:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{antic_et_al:LIPIcs.GD.2025.47,
  author =	{Anti\'{c}, Todor and Gamboa Quintero, Guillermo and Gli\v{s}i\'{c}, Jelena},
  title =	{{Reconfigurations of Plane Caterpillars and Paths}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{47:1--47:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.47},
  URN =		{urn:nbn:de:0030-drops-250337},
  doi =		{10.4230/LIPIcs.GD.2025.47},
  annote =	{Keywords: reconfiguration graph, caterpillar, path, geometric graph}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.35

A Walk on the Wild Side: A Shape-First Methodology for Orthogonal Drawings

Authors: Giordano Andreola, Susanna Caroppo, Giuseppe Di Battista, Fabrizio Grosso, Maurizio Patrignani, and Allegra Strippoli

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

Several algorithms for the construction of orthogonal drawings of graphs, including those based on the Topology-Shape-Metrics (TSM) paradigm, tend to prioritize the minimization of crossings. This emphasis has two notable side effects: some edges are drawn with unnecessarily long sequences of segments and bends, and the overall drawing area may become excessively large. As a result, the produced drawings often lack geometric uniformity. Moreover, orthogonal crossings are known to have a limited impact on readability, suggesting that crossing minimization may not always be the optimal goal. In this paper, we introduce a methodology that "subverts" the traditional TSM pipeline by focusing on minimizing bends. Given a graph G, we ideally seek to construct a rectilinear drawing of G, that is, an orthogonal drawing with no bends. When not possible, we incrementally subdivide the edges of G by introducing dummy vertices that will (possibly) correspond to bends in the final drawing. This process continues until a rectilinear drawing of a subdivision of the graph is found, after which the final coordinates are computed. We tackle the (NP-complete) rectilinear drawability problem by encoding it as a SAT formula and solving it with state-of-the-art SAT solvers. If the SAT formula is unsatisfiable, we use the solver’s proof to determine which edge to subdivide. Our implementation, domus, which is fairly simple, is evaluated through extensive experiments on small- to medium-sized graphs. The results show that it consistently outperforms ogdf’s TSM-based approach across most standard graph drawing metrics.

Cite as

Giordano Andreola, Susanna Caroppo, Giuseppe Di Battista, Fabrizio Grosso, Maurizio Patrignani, and Allegra Strippoli. A Walk on the Wild Side: A Shape-First Methodology for Orthogonal Drawings. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{andreola_et_al:LIPIcs.GD.2025.35,
  author =	{Andreola, Giordano and Caroppo, Susanna and Di Battista, Giuseppe and Grosso, Fabrizio and Patrignani, Maurizio and Strippoli, Allegra},
  title =	{{A Walk on the Wild Side: A Shape-First Methodology for Orthogonal Drawings}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{35:1--35:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.35},
  URN =		{urn:nbn:de:0030-drops-250218},
  doi =		{10.4230/LIPIcs.GD.2025.35},
  annote =	{Keywords: Non-planar Orthogonal Drawings, SAT Solver, Experimental Comparison}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.34

Layered Polyline Drawings of Planar Graphs

Authors: Debajyoti Mondal

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

A k-layer polyline drawing of a planar graph G is a planar drawing of G on a set L of k parallel lines such that each vertex is mapped to a point on L and each edge is mapped to a polygonal chain with the endpoints and bends lying on L. In the fixed embedding setting, the output drawing maintains the given planar embedding, whereas in the variable embedding setting, the embedding may change. Every n-vertex planar graph admits a polyline drawing on 2n/3 layers, which is the best known upper bound for both settings. We improve this bound in the variable embedding setting. We show that every planar graph can be drawn on 14n/27+O(√n) layers by choosing a proper planar embedding, breaking the long-standing 2n/3-layer barrier.

Cite as

Debajyoti Mondal. Layered Polyline Drawings of Planar Graphs. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 34:1-34:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mondal:LIPIcs.GD.2025.34,
  author =	{Mondal, Debajyoti},
  title =	{{Layered Polyline Drawings of Planar Graphs}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{34:1--34:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.34},
  URN =		{urn:nbn:de:0030-drops-250202},
  doi =		{10.4230/LIPIcs.GD.2025.34},
  annote =	{Keywords: Layered Drawing, Variable Embedding, Polyline Drawing, Cycle Separator}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.111

Hardness of Median and Center in the Ulam Metric

Authors: Nick Fischer, Elazar Goldenberg, Mursalin Habib, and Karthik C. S.

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. - Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. - Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive Õ(n² L)-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

Cite as

Nick Fischer, Elazar Goldenberg, Mursalin Habib, and Karthik C. S.. Hardness of Median and Center in the Ulam Metric. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 111:1-111:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fischer_et_al:LIPIcs.ESA.2025.111,
  author =	{Fischer, Nick and Goldenberg, Elazar and Habib, Mursalin and Karthik C. S.},
  title =	{{Hardness of Median and Center in the Ulam Metric}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{111:1--111:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.111},
  URN =		{urn:nbn:de:0030-drops-245809},
  doi =		{10.4230/LIPIcs.ESA.2025.111},
  annote =	{Keywords: Ulam distance, median, center, rank aggregation, fine-grained complexity}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.90

Separating Two Points with Obstacles in the Plane: Improved Upper and Lower Bounds

Authors: Jack Spalding-Jamieson and Anurag Murty Naredla

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Given two points in the plane, and a set of "obstacles" given as curves through the plane with assigned weights, we consider the point-separation problem, which asks for a minimum-weight subset of the obstacles separating the two points. A few computational models for this problem have been previously studied. We give a unified approach to this problem in all models via a reduction to a particular shortest-path problem, and obtain improved running times in essentially all cases. In addition, we also give fine-grained lower bounds for many cases.

Cite as

Jack Spalding-Jamieson and Anurag Murty Naredla. Separating Two Points with Obstacles in the Plane: Improved Upper and Lower Bounds. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 90:1-90:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{spaldingjamieson_et_al:LIPIcs.ESA.2025.90,
  author =	{Spalding-Jamieson, Jack and Naredla, Anurag Murty},
  title =	{{Separating Two Points with Obstacles in the Plane: Improved Upper and Lower Bounds}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{90:1--90:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.90},
  URN =		{urn:nbn:de:0030-drops-245598},
  doi =		{10.4230/LIPIcs.ESA.2025.90},
  annote =	{Keywords: obstacle separation, point separation, geometric intersection graph, Z₂-homology, fine-grained lower bounds}
}

Document

APPROX

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.10

Improved Approximation Guarantees for Advertisement Placement

Authors: Waldo Gálvez, Roberto Oliva, and Victor Verdugo

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract

The advertisement placement problem involves selecting and scheduling ads within a timeline that has capacity constraints to maximize profit. Each task is characterized by its height, width, and profit, and must be fully scheduled across multiple time slots. This problem models practical scenarios such as internet advertising and energy management, and it also generalizes classical combinatorial optimization problems like the knapsack and bin packing problems. We present a simple (2+ε)-approximation algorithm for any ε > 0, which improves upon the state-of-the-art 3+ε factor established by Freund and Naor twenty years ago. Our approach combines rounding techniques with dynamic programming and an efficient extension of list scheduling. Furthermore, we enhance this method with linear programming techniques to provide an almost optimal (1+ε)-approximation algorithm under resource augmentation, which allows for a slight increase in time slot capacities.

Cite as

Waldo Gálvez, Roberto Oliva, and Victor Verdugo. Improved Approximation Guarantees for Advertisement Placement. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{galvez_et_al:LIPIcs.APPROX/RANDOM.2025.10,
  author =	{G\'{a}lvez, Waldo and Oliva, Roberto and Verdugo, Victor},
  title =	{{Improved Approximation Guarantees for Advertisement Placement}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.10},
  URN =		{urn:nbn:de:0030-drops-243762},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.10},
  annote =	{Keywords: Advertisement Placement, Two-dimensional Packing, Geometric Knapsack, Resource Allocation}
}

Document

DOI: 10.4230/LIPIcs.WADS.2025.5

On Planar Straight-Line Dominance Drawings

Authors: Patrizio Angelini, Michael A. Bekos, Giuseppe Di Battista, Fabrizio Frati, Luca Grilli, and Giacomo Ortali

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract

We study the following question, which has been considered since the 90’s: Does every st-planar graph admit a planar straight-line dominance drawing? We show concrete evidence for the difficulty of this question, by proving that, unlike upward planar straight-line drawings, planar straight-line dominance drawings with prescribed y-coordinates do not always exist and planar straight-line dominance drawings cannot always be constructed via a contract-draw-expand inductive approach. We also show several classes of st-planar graphs that always admit a planar straight-line dominance drawing. These include st-planar 3-trees in which every stacking operation introduces two edges incoming into the new vertex, st-planar graphs in which every vertex is adjacent to the sink, and st-planar graphs in which no face has the left boundary that is a single edge.

Cite as

Patrizio Angelini, Michael A. Bekos, Giuseppe Di Battista, Fabrizio Frati, Luca Grilli, and Giacomo Ortali. On Planar Straight-Line Dominance Drawings. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{angelini_et_al:LIPIcs.WADS.2025.5,
  author =	{Angelini, Patrizio and Bekos, Michael A. and Di Battista, Giuseppe and Frati, Fabrizio and Grilli, Luca and Ortali, Giacomo},
  title =	{{On Planar Straight-Line Dominance Drawings}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.5},
  URN =		{urn:nbn:de:0030-drops-242361},
  doi =		{10.4230/LIPIcs.WADS.2025.5},
  annote =	{Keywords: st-graphs, dominance drawings, planar straight-line drawings, upward planarity}
}

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