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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

The minimum genus of a graph is an important question in graph theory and a key ingredient in several graph algorithms. However, its computation is NP-hard and turns out to be hard even in practice. Only recently, the first non-trivial approach - based on SAT and ILP (integer linear programming) models - has been presented, but it is unable to successfully tackle graphs of genus larger than 1 in practice.
Herein, we show how to improve the ILP formulation. The crucial ingredients are two-fold. First, we show that instead of modeling rotation schemes explicitly, it suffices to optimize over partitions of the (bidirected) arc set A of the graph. Second, we exploit the cycle structure of the graph, explicitly mapping short closed walks on A to faces in the embedding.
Besides the theoretical advantages of our models, we show their practical strength by a thorough experimental evaluation. Contrary to the previous approach, we are able to quickly solve many instances of genus > 1.

Markus Chimani and Tilo Wiedera. Stronger ILPs for the Graph Genus Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chimani_et_al:LIPIcs.ESA.2019.30, author = {Chimani, Markus and Wiedera, Tilo}, title = {{Stronger ILPs for the Graph Genus Problem}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {30:1--30:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.30}, URN = {urn:nbn:de:0030-drops-111511}, doi = {10.4230/LIPIcs.ESA.2019.30}, annote = {Keywords: algorithm engineering, genus, integer linear programming} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c >= 13 and d >= 1, we give first explicit constructions of c-crossing-critical graphs containing a vertex of degree greater than d. We also show that such unbounded degree constructions do not exist for c <=12, precisely, that there exists a constant D such that every c-crossing-critical graph with c <=12 has maximum degree at most D. Hence, the bounded maximum degree conjecture of c-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c <=12.

Drago Bokal, Zdeněk Dvořák, Petr Hliněný, Jesús Leaños, Bojan Mohar, and Tilo Wiedera. Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bokal_et_al:LIPIcs.SoCG.2019.14, author = {Bokal, Drago and Dvo\v{r}\'{a}k, Zden\v{e}k and Hlin\v{e}n\'{y}, Petr and Lea\~{n}os, Jes\'{u}s and Mohar, Bojan and Wiedera, Tilo}, title = {{Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c \langle= 12}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.14}, URN = {urn:nbn:de:0030-drops-104183}, doi = {10.4230/LIPIcs.SoCG.2019.14}, annote = {Keywords: graph drawing, crossing number, crossing-critical, zip product} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

The NP-hard Maximum Planar Subgraph problem asks for a planar subgraph H of a given graph G such that H has maximum edge cardinality. For more than two decades, the only known non-trivial exact algorithm was based on integer linear programming and Kuratowski's famous planarity criterion. We build upon this approach and present new constraint classes - together with a lifting of the polyhedron - to obtain provably stronger LP-relaxations, and in turn faster algorithms in practice. The new constraints take Euler's polyhedron formula as a starting point and combine it with considering cycles in G. This paper discusses both the theoretical as well as the practical sides of this strengthening.

Markus Chimani and Tilo Wiedera. Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chimani_et_al:LIPIcs.ESA.2018.19, author = {Chimani, Markus and Wiedera, Tilo}, title = {{Cycles to the Rescue! Novel Constraints to Compute Maximum Planar Subgraphs Fast}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.19}, URN = {urn:nbn:de:0030-drops-94829}, doi = {10.4230/LIPIcs.ESA.2018.19}, annote = {Keywords: algorithm engineering, graph algorithms, integer linear programming, maximum planar subgraph} }

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**Published in:** LIPIcs, Volume 103, 17th International Symposium on Experimental Algorithms (SEA 2018)

Given a graph G, the NP-hard Maximum Planar Subgraph problem asks for a planar subgraph of G with the maximum number of edges. The only known non-trivial exact algorithm utilizes Kuratowski's famous planarity criterion and can be formulated as an integer linear program (ILP) or a pseudo-boolean satisfiability problem (PBS). We examine three alternative characterizations of planarity regarding their applicability to model maximum planar subgraphs. For each, we consider both ILP and PBS variants, investigate diverse formulation aspects, and evaluate their practical performance.

Markus Chimani, Ivo Hedtke, and Tilo Wiedera. Exact Algorithms for the Maximum Planar Subgraph Problem: New Models and Experiments. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chimani_et_al:LIPIcs.SEA.2018.22, author = {Chimani, Markus and Hedtke, Ivo and Wiedera, Tilo}, title = {{Exact Algorithms for the Maximum Planar Subgraph Problem: New Models and Experiments}}, booktitle = {17th International Symposium on Experimental Algorithms (SEA 2018)}, pages = {22:1--22:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-070-5}, ISSN = {1868-8969}, year = {2018}, volume = {103}, editor = {D'Angelo, Gianlorenzo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2018.22}, URN = {urn:nbn:de:0030-drops-89572}, doi = {10.4230/LIPIcs.SEA.2018.22}, annote = {Keywords: maximum planar subgraph, integer linear programming, pseudo boolean satisfiability, graph drawing, algorithm engineering} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Formally, approaches based on mathematical programming are able to find provably optimal solutions.
However, the demands on a verifiable formal proof are typically much higher than the guarantees
we can sensibly attribute to implementations of mathematical programs. We consider this in the context of the crossing number problem, one of the most prominent problems in topological graph theory. The problem asks for the minimum number of edge crossings in any drawing of a given graph. Graph-theoretic proofs for this problem are known to be notoriously hard to obtain. At the same time, proofs even for very specific graphs are often of interest in crossing number research, as they can, e.g., form the basis for inductive proofs.
We propose a system to automatically generate a formal proof based on an ILP computation. Such a proof is (relatively) easily verifiable, and does not require the understanding of any complex ILP codes. As such, we hope our proof system may serve as a showcase for the necessary steps and central design goals of how to establish formal proof systems based on mathematical programming formulations.

Markus Chimani and Tilo Wiedera. An ILP-based Proof System for the Crossing Number Problem. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chimani_et_al:LIPIcs.ESA.2016.29, author = {Chimani, Markus and Wiedera, Tilo}, title = {{An ILP-based Proof System for the Crossing Number Problem}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {29:1--29:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.29}, URN = {urn:nbn:de:0030-drops-63803}, doi = {10.4230/LIPIcs.ESA.2016.29}, annote = {Keywords: automatic formal proof, crossing number, integer linear programming} }

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