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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Two graphs are homomorphism indistinguishable over a graph class 𝐅, denoted by G ≡_𝐅 H, if hom(F,G) = hom(F,H) for all F ∈ 𝐅 where hom(F,G) denotes the number of homomorphisms from F to G. A classical result of Lovász shows that isomorphism between graphs is equivalent to homomorphism indistinguishability over the class of all graphs. More recently, there has been a series of works giving natural algebraic and/or logical characterizations for homomorphism indistinguishability over certain restricted graph classes.
A class of graphs 𝐅 is homomorphism-distinguishing closed if, for every F ∉ 𝐅, there are graphs G and H such that G ≡_𝐅 H and hom(F,G) ≠ hom(F,H). Roberson conjectured that every class closed under taking minors and disjoint unions is homomorphism-distinguishing closed which implies that every such class defines a distinct equivalence relation between graphs. In this work, we confirm this conjecture for the classes 𝒯_k, k ≥ 1, containing all graphs of tree-width at most k.
As an application of this result, we also characterize which subgraph counts are detected by the k-dimensional Weisfeiler-Leman algorithm. This answers an open question from [Arvind et al., J. Comput. Syst. Sci., 2020].

Daniel Neuen. Homomorphism-Distinguishing Closedness for Graphs of Bounded Tree-Width. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{neuen:LIPIcs.STACS.2024.53, author = {Neuen, Daniel}, title = {{Homomorphism-Distinguishing Closedness for Graphs of Bounded Tree-Width}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {53:1--53:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.53}, URN = {urn:nbn:de:0030-drops-197630}, doi = {10.4230/LIPIcs.STACS.2024.53}, annote = {Keywords: homomorphism indistinguishability, tree-width, Weisfeiler-Leman algorithm, subgraph counts} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

In a recent work, Esmer et al. describe a simple method - Approximate Monotone Local Search - to obtain exponential approximation algorithms from existing parameterized exact algorithms, polynomial-time approximation algorithms and, more generally, parameterized approximation algorithms. In this work, we generalize those results to the weighted setting.
More formally, we consider monotone subset minimization problems over a weighted universe of size n (e.g., Vertex Cover, d-Hitting Set and Feedback Vertex Set). We consider a model where the algorithm is only given access to a subroutine that finds a solution of weight at most α ⋅ W (and of arbitrary cardinality) in time c^k ⋅ n^{𝒪(1)} where W is the minimum weight of a solution of cardinality at most k. In the unweighted setting, Esmer et al. determine the smallest value d for which a β-approximation algorithm running in time dⁿ ⋅ n^{𝒪(1)} can be obtained in this model. We show that the same dependencies also hold in a weighted setting in this model: for every fixed ε > 0 we obtain a β-approximation algorithm running in time 𝒪((d+ε)ⁿ), for the same d as in the unweighted setting.
Similarly, we also extend a β-approximate brute-force search (in a model which only provides access to a membership oracle) to the weighted setting. Using existing approximation algorithms and exact parameterized algorithms for weighted problems, we obtain the first exponential-time β-approximation algorithms that are better than brute force for a variety of problems including Weighted Vertex Cover, Weighted d-Hitting Set, Weighted Feedback Vertex Set and Weighted Multicut.

Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Approximate Monotone Local Search for Weighted Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 17:1-17:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{esmer_et_al:LIPIcs.IPEC.2023.17, author = {Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Neuen, Daniel and Sharma, Roohani}, title = {{Approximate Monotone Local Search for Weighted Problems}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {17:1--17:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.17}, URN = {urn:nbn:de:0030-drops-194360}, doi = {10.4230/LIPIcs.IPEC.2023.17}, annote = {Keywords: parameterized approximations, exponential approximations, monotone local search} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized α-approximation algorithm that runs in c^k⋅n^𝒪(1) time, where k is the solution size, can be used to derive an α-approximation randomized algorithm that runs in dⁿ⋅n^𝒪(1) time, where d is the unique value in (1, 1+{c-1}/α) such that 𝒟(1/α‖{d-1}/{c-1}) = {ln c}/α and 𝒟(a‖b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for α = 1, and is strictly better when α > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time.
We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2ⁿ⋅n^𝒪(1) exhaustive search can be adapted to an α-approximate exhaustive search that runs in time (1+exp(-α⋅ℋ(1/(α))))ⁿ⋅n^𝒪(1), where ℋ is the entropy function. Furthermore, we provide a lower bound stating that the running time of this α-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any α ≥ 1, c > 1.
We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114ⁿ⋅n^𝒪(1), improving upon the previously best known 1.1-approximation running in time 1.127ⁿ⋅n^𝒪(1) by Bourgeois et al. [DAM 2011].

Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{esmer_et_al:LIPIcs.ESA.2022.50, author = {Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Neuen, Daniel and Sharma, Roohani}, title = {{Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {50:1--50:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.50}, URN = {urn:nbn:de:0030-drops-169887}, doi = {10.4230/LIPIcs.ESA.2022.50}, annote = {Keywords: parameterized approximations, exponential approximations, monotone local search} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

The Weisfeiler-Leman (WL) algorithm is a combinatorial procedure that computes colorings on graphs, which can often be used to detect their (non-)isomorphism. Particularly the 1- and 2-dimensional versions 1-WL and 2-WL have received much attention, due to their numerous links to other areas of computer science.
Knowing the expressive power of a certain dimension of the algorithm usually amounts to understanding the computed colorings. An increase in the dimension leads to finer computed colorings and, thus, more graphs can be distinguished. For example, on the class of planar graphs, 3-WL solves the isomorphism problem. However, the expressive power of 2-WL on the class is poorly understood (and, in particular, it may even well be that it decides isomorphism).
In this paper, we investigate the colorings computed by 2-WL on planar graphs. Towards this end, we analyze the graphs induced by edge color classes in the graph. Based on the obtained classification, we show that for every 3-connected planar graph, it holds that: a) after coloring all pairs with their 2-WL color, the graph has fixing number 1 with respect to 1-WL, or b) there is a 2-WL-definable matching that can be used to transform the graph into a smaller one, or c) 2-WL detects a connected subgraph that is essentially the graph of a Platonic or Archimedean solid, a prism, a cycle, or a bipartite graph K_{2,𝓁}. In particular, the graphs from case (a) are identified by 2-WL.

Sandra Kiefer and Daniel Neuen. A Study of Weisfeiler-Leman Colorings on Planar Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 81:1-81:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kiefer_et_al:LIPIcs.ICALP.2022.81, author = {Kiefer, Sandra and Neuen, Daniel}, title = {{A Study of Weisfeiler-Leman Colorings on Planar Graphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {81:1--81:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.81}, URN = {urn:nbn:de:0030-drops-164228}, doi = {10.4230/LIPIcs.ICALP.2022.81}, annote = {Keywords: Weisfeiler-Leman algorithm, planar graphs, edge-transitive graphs, fixing number} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We present an isomorphism test for graphs of Euler genus g running in time 2^{{O}(g⁴ log g)}n^{{O}(1)}. Our algorithm provides the first explicit upper bound on the dependence on g for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time f(g)n for some function f (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude K_{3,h} as a minor. For such graphs, no fpt isomorphism test was known before.
The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, we introduce (t,k)-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm. This concept may be of independent interest.

Daniel Neuen. Isomorphism Testing Parameterized by Genus and Beyond. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 72:1-72:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{neuen:LIPIcs.ESA.2021.72, author = {Neuen, Daniel}, title = {{Isomorphism Testing Parameterized by Genus and Beyond}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {72:1--72:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.72}, URN = {urn:nbn:de:0030-drops-146533}, doi = {10.4230/LIPIcs.ESA.2021.72}, annote = {Keywords: graph isomorphism, fixed-parameter tractability, Euler genus, Weisfeiler-Leman algorithm} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices V and a permutation group Γ over domain V, and asking whether there is a permutation γ ∈ Γ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on d points, this problem can be solved in time (n+m)^O((log d)^c) for some absolute constant c where n denotes the number of vertices and m the number of hyperedges. In particular, this gives the currently fastest isomorphism test for hypergraphs in general. The previous best algorithm for the above problem due to Schweitzer and Wiebking (STOC 2019) runs in time n^O(d)m^O(1).
As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding K_{3,h} as a minor in time n^O((log h)^c). In particular, this gives an isomorphism test for graphs of Euler genus at most g running in time n^O((log g)^c).

Daniel Neuen. Hypergraph Isomorphism for Groups with Restricted Composition Factors. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 88:1-88:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{neuen:LIPIcs.ICALP.2020.88, author = {Neuen, Daniel}, title = {{Hypergraph Isomorphism for Groups with Restricted Composition Factors}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {88:1--88:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.88}, URN = {urn:nbn:de:0030-drops-124959}, doi = {10.4230/LIPIcs.ICALP.2020.88}, annote = {Keywords: graph isomorphism, groups with restricted composition factors, hypergraphs, bounded genus graphs} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into bi- and triconnected components.
We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its triconnected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3-connected components (assuming dimension at least 2).
This result implies that for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes.
In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a construction by Cai, Fürer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.

Sandra Kiefer and Daniel Neuen. The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kiefer_et_al:LIPIcs.MFCS.2019.45, author = {Kiefer, Sandra and Neuen, Daniel}, title = {{The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {45:1--45:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.45}, URN = {urn:nbn:de:0030-drops-109893}, doi = {10.4230/LIPIcs.MFCS.2019.45}, annote = {Keywords: Weisfeiler-Leman, separators, first-order logic, counting quantifiers} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We give a new fpt algorithm testing isomorphism of n-vertex graphs of tree width k in time 2^{k polylog(k)} poly n, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2^{O(k^5 log k)}poly n. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width k. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai's algorithm as a black box in one place. We give a second algorithm which, at the price of a slightly worse run time 2^{O(k^2 log k)}poly n, avoids the use of Babai's algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm.

Martin Grohe, Daniel Neuen, Pascal Schweitzer, and Daniel Wiebking. An Improved Isomorphism Test for Bounded-Tree-Width Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 67:1-67:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{grohe_et_al:LIPIcs.ICALP.2018.67, author = {Grohe, Martin and Neuen, Daniel and Schweitzer, Pascal and Wiebking, Daniel}, title = {{An Improved Isomorphism Test for Bounded-Tree-Width Graphs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {67:1--67:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.67}, URN = {urn:nbn:de:0030-drops-90714}, doi = {10.4230/LIPIcs.ICALP.2018.67}, annote = {Keywords: graph isomorphism, graph canonization, tree width, decompositions} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

The state-of-the-art solvers for the graph isomorphism problem can readily solve generic instances with tens of thousands of vertices. Indeed, experiments show that on inputs without particular combinatorial structure the algorithms scale almost linearly. In fact, it is non-trivial to create challenging instances for such solvers and the number of difficult benchmark graphs available is quite limited.
We describe a construction to efficiently generate small instances for the graph isomorphism problem that are difficult or even infeasible for said solvers.
Up to this point the only other available instances posing challenges for isomorphism solvers were certain incidence structures of combinatorial objects (such as projective planes, Hadamard matrices, Latin squares, etc.). Experiments show that starting from 1500 vertices our new instances are several orders of magnitude more difficult on comparable input sizes. More importantly, our method is generic and efficient in the sense that one can quickly create many isomorphism instances on a desired number of vertices. In contrast to this, said combinatorial objects are rare and difficult to generate and with the new construction it is possible to generate an abundance of instances of arbitrary size.
Our construction hinges on the multipedes of Gurevich and Shelah and the Cai-Fürer-Immerman gadgets that realize a certain abelian automorphism group and have repeatedly played a role in the context of graph isomorphism. Exploring limits, we also explain that there are group theoretic obstructions to generalizing the construction with non-abelian gadgets.

Daniel Neuen and Pascal Schweitzer. Benchmark Graphs for Practical Graph Isomorphism. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{neuen_et_al:LIPIcs.ESA.2017.60, author = {Neuen, Daniel and Schweitzer, Pascal}, title = {{Benchmark Graphs for Practical Graph Isomorphism}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {60:1--60:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.60}, URN = {urn:nbn:de:0030-drops-78701}, doi = {10.4230/LIPIcs.ESA.2017.60}, annote = {Keywords: graph isomorphism, benchmark instances, practical solvers} }

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

In the past decades for more and more graph classes the Graph Isomorphism Problem was shown to be solvable in polynomial time. An interesting family of graph classes arises from intersection graphs of geometric objects. In this work we show that the Graph Isomorphism Problem for unit square graphs, intersection graphs of axis-parallel unit squares in the plane, can be solved in polynomial time. Since the recognition problem for this class of graphs is NP-hard we can not rely on standard techniques for geometric graphs based on constructing a canonical realization. Instead, we develop new techniques which combine structural insights into the class of unit square graphs with understanding of the automorphism group of such graphs. For the latter we introduce a generalization of bounded degree graphs which is used to capture the main structure of unit square graphs. Using group theoretic algorithms we obtain sufficient information to solve the isomorphism problem for unit square graphs.

Daniel Neuen. Graph Isomorphism for Unit Square Graphs. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{neuen:LIPIcs.ESA.2016.70, author = {Neuen, Daniel}, title = {{Graph Isomorphism for Unit Square Graphs}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {70:1--70:17}, 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.2016.70}, URN = {urn:nbn:de:0030-drops-64120}, doi = {10.4230/LIPIcs.ESA.2016.70}, annote = {Keywords: graph isomorphism, geometric graphs, unit squares} }

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