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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g ≥ 1 and recovers the known tight bound for the planar case (g = 0).

Júlia Baligács, Yann Disser, Irene Heinrich, and Pascal Schweitzer. Exploration of Graphs with Excluded Minors. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{baligacs_et_al:LIPIcs.ESA.2023.11, author = {Balig\'{a}cs, J\'{u}lia and Disser, Yann and Heinrich, Irene and Schweitzer, Pascal}, title = {{Exploration of Graphs with Excluded Minors}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {11:1--11:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.11}, URN = {urn:nbn:de:0030-drops-186644}, doi = {10.4230/LIPIcs.ESA.2023.11}, annote = {Keywords: online algorithms, competitive analysis, graph exploration, graph spanners, minor-free graphs, bounded genus graphs} }

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**Published in:** LIPIcs, Volume 271, 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)

Dedicated treatment of symmetries in satisfiability problems (SAT) is indispensable for solving various classes of instances arising in practice. However, the exploitation of symmetries usually takes a black box approach. Typically, off-the-shelf external, general-purpose symmetry detection tools are invoked to compute symmetry groups of a formula. The groups thus generated are a set of permutations passed to a separate tool to perform further analyzes to understand the structure of the groups. The result of this second computation is in turn used for tasks such as static symmetry breaking or dynamic pruning of the search space. Within this pipeline of tools, the detection and analysis of symmetries typically incurs the majority of the time overhead for symmetry exploitation.
In this paper we advocate for a more holistic view of what we call the SAT-symmetry interface. We formulate a computational setting, centered around a new concept of joint graph/group pairs, to analyze and improve the detection and analysis of symmetries. Using our methods, no information is lost performing computational tasks lying on the SAT-symmetry interface. Having access to the entire input allows for simpler, yet efficient algorithms.
Specifically, we devise algorithms and heuristics for computing finest direct disjoint decompositions, finding equivalent orbits, and finding natural symmetric group actions. Our algorithms run in what we call instance-quasi-linear time, i.e., almost linear time in terms of the input size of the original formula and the description length of the symmetry group returned by symmetry detection tools. Our algorithms improve over both heuristics used in state-of-the-art symmetry exploitation tools, as well as theoretical general-purpose algorithms.

Markus Anders, Pascal Schweitzer, and Mate Soos. Algorithms Transcending the SAT-Symmetry Interface. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{anders_et_al:LIPIcs.SAT.2023.1, author = {Anders, Markus and Schweitzer, Pascal and Soos, Mate}, title = {{Algorithms Transcending the SAT-Symmetry Interface}}, booktitle = {26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)}, pages = {1:1--1:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-286-0}, ISSN = {1868-8969}, year = {2023}, volume = {271}, editor = {Mahajan, Meena and Slivovsky, Friedrich}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2023.1}, URN = {urn:nbn:de:0030-drops-184635}, doi = {10.4230/LIPIcs.SAT.2023.1}, annote = {Keywords: boolean satisfiability, symmetry exploitation, computational group theory} }

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**Published in:** LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

State-of-the-art solvers for symmetry detection in combinatorial objects are becoming increasingly sophisticated software libraries. Most of the solvers were initially designed with inputs from combinatorics in mind (nauty, bliss, Traces, dejavu). They excel at dealing with a complicated core of the input. Others focus on practical instances that exhibit sparsity. They excel at dealing with comparatively easy but extremely large substructures of the input (saucy). In practice, these differences manifest in significantly diverging performances on different types of graph classes.
We engineer a preprocessor for symmetry detection. The result is a tool designed to shrink sparse, large substructures of the input graph. On most of the practical instances, the preprocessor improves the overall running time significantly for many of the state-of-the-art solvers. At the same time, our benchmarks show that the additional overhead is negligible.
Overall we obtain single algorithms with competitive performance across all benchmark graphs. As such, the preprocessor bridges the disparity between solvers that focus on combinatorial graphs and large practical graphs. In fact, on most of the practical instances the combined setup significantly outperforms previous state-of-the-art.

Markus Anders, Pascal Schweitzer, and Julian Stieß. Engineering a Preprocessor for Symmetry Detection. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{anders_et_al:LIPIcs.SEA.2023.1, author = {Anders, Markus and Schweitzer, Pascal and Stie{\ss}, Julian}, title = {{Engineering a Preprocessor for Symmetry Detection}}, booktitle = {21st International Symposium on Experimental Algorithms (SEA 2023)}, pages = {1:1--1:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-279-2}, ISSN = {1868-8969}, year = {2023}, volume = {265}, editor = {Georgiadis, Loukas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.1}, URN = {urn:nbn:de:0030-drops-183511}, doi = {10.4230/LIPIcs.SEA.2023.1}, annote = {Keywords: graph isomorphism, automorphism groups, symmetry detection, preprocessors} }

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

We investigate the relationship between various isomorphism invariants for finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to characterize, compare and quantify the effectiveness and complexity of invariants for group isomorphism.
It turns out that a surprising number of invariants and characteristic subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm. These include the center, the inner automorphism group, the commutator subgroup and the derived series, the abelian radical, the solvable radical, the Fitting group and π-radicals. A low dimensional WL-algorithm additionally determines the isomorphism type of the socle as well as the factors in the derives series and the upper and lower central series.
We also analyze the behavior of the WL-algorithm for group extensions and prove that a low dimensional WL-algorithm determines the isomorphism types of the composition factors of a group.
Finally we develop a new tool to define a canonical maximal central decomposition for groups. This allows us to show that the Weisfeiler-Leman dimension of a group is at most one larger than the dimensions of its direct indecomposable factors. In other words the Weisfeiler-Leman dimension increases by at most 1 when taking direct products.

Jendrik Brachter and Pascal Schweitzer. A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{brachter_et_al:LIPIcs.ESA.2022.27, author = {Brachter, Jendrik and Schweitzer, Pascal}, title = {{A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {27:1--27:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.27}, URN = {urn:nbn:de:0030-drops-169653}, doi = {10.4230/LIPIcs.ESA.2022.27}, annote = {Keywords: group isomorphism problem, Weisfeiler-Leman algorithms, group invariants, direct product decompositions} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

Individualization-Refinement (IR) algorithms form the standard method and currently the only practical method for symmetry computations of graphs and combinatorial objects in general. Through backtracking, on each graph an IR-algorithm implicitly creates an IR-tree whose order is the determining factor of the running time of the algorithm.
We give a precise and constructive characterization which trees are IR-trees. This characterization is applicable both when the tree is regarded as an uncolored object but also when regarded as a colored object where vertex colors stem from a node invariant. We also provide a construction that given a tree produces a corresponding graph whenever possible. This provides a constructive proof that our necessary conditions are also sufficient for the characterization.

Markus Anders, Jendrik Brachter, and Pascal Schweitzer. A Characterization of Individualization-Refinement Trees. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anders_et_al:LIPIcs.ISAAC.2021.24, author = {Anders, Markus and Brachter, Jendrik and Schweitzer, Pascal}, title = {{A Characterization of Individualization-Refinement Trees}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.24}, URN = {urn:nbn:de:0030-drops-154578}, doi = {10.4230/LIPIcs.ISAAC.2021.24}, annote = {Keywords: individualization refinement algorithms, backtracking trees, graph isomorphism} }

Document

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

In practice symmetries of combinatorial structures are computed by transforming the structure into an annotated graph whose automorphisms correspond exactly to the desired symmetries. An automorphism solver is then employed to compute the automorphism group of the constructed graph. Such solvers have been developed for over 50 years, and highly efficient sequential, single core tools are available. However no competitive parallel tools are available for the task.
We introduce a new parallel randomized algorithm that is based on a modification of the individualization-refinement paradigm used by sequential solvers. The use of randomization crucially enables parallelization.
We report extensive benchmark results that show that our solver is competitive to state-of-the-art solvers on a single thread, while scaling remarkably well with the use of more threads. This results in order-of-magnitude improvements on many graph classes over state-of-the-art solvers. In fact, our tool is the first parallel graph automorphism tool that outperforms current sequential tools.

Markus Anders and Pascal Schweitzer. Parallel Computation of Combinatorial Symmetries. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anders_et_al:LIPIcs.ESA.2021.6, author = {Anders, Markus and Schweitzer, Pascal}, title = {{Parallel Computation of Combinatorial Symmetries}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {6:1--6:18}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.6}, URN = {urn:nbn:de:0030-drops-145875}, doi = {10.4230/LIPIcs.ESA.2021.6}, annote = {Keywords: graph isomorphism, automorphism groups, algorithm engineering, parallel algorithms} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Color refinement is a crucial subroutine in symmetry detection in theory as well as practice. It has further applications in machine learning and in computational problems from linear algebra.
While tight lower bounds for the worst case complexity are known [Berkholz, Bonsma, Grohe, ESA2013] no comparative analysis of design choices for color refinement algorithms is available.
We devise two models within which we can compare color refinement algorithms using formal methods, an online model and an approximation model. We use these to show that no online algorithm is competitive beyond a logarithmic factor and no algorithm can approximate the optimal color refinement splitting scheme beyond a logarithmic factor.
We also directly compare strategies used in practice showing that, on some graphs, queue based strategies outperform stack based ones by a logarithmic factor and vice versa. Similar results hold for strategies based on priority queues.

Markus Anders, Pascal Schweitzer, and Florian Wetzels. Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anders_et_al:LIPIcs.ICALP.2021.15, author = {Anders, Markus and Schweitzer, Pascal and Wetzels, Florian}, title = {{Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.15}, URN = {urn:nbn:de:0030-drops-140846}, doi = {10.4230/LIPIcs.ICALP.2021.15}, annote = {Keywords: Color refinement, Online algorithms, Graph isomorphism, Lower bounds} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We define a search problem on trees that closely captures the backtracking behavior of all current practical graph isomorphism algorithms. Given two trees with colored leaves, the goal is to find two leaves of matching color, one in each of the trees. The trees are subject to an invariance property which promises that for every pair of leaves of equal color there must be a symmetry (or an isomorphism) that maps one leaf to the other.
We describe a randomized algorithm with errors for which the number of visited nodes is quasilinear in the square root of the size of the smaller of the two trees. For inputs of bounded degree, we develop a Las Vegas algorithm with a similar running time.
We prove that these results are optimal up to logarithmic factors. For this, we show a lower bound for randomized algorithms on inputs of bounded degree that is the square root of the tree sizes. For inputs of unbounded degree, we show a linear lower bound for Las Vegas algorithms. For deterministic algorithms we can prove a linear bound even for inputs of bounded degree. This shows why randomized algorithms outperform deterministic ones.
Our results explain why the randomized "breadth-first with intermixed experimental path" search strategy of the isomorphism tool Traces (Piperno 2008) is often superior to the depth-first search strategy of other tools such as nauty (McKay 1977) or bliss (Junttila, Kaski 2007). However, our algorithm also provides a new traversal strategy, which is theoretically near optimal and which has better worst case behavior than traversal strategies that have previously been used.

Markus Anders and Pascal Schweitzer. Search Problems in Trees with Symmetries: Near Optimal Traversal Strategies for Individualization-Refinement Algorithms. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 16:1-16:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{anders_et_al:LIPIcs.ICALP.2021.16, author = {Anders, Markus and Schweitzer, Pascal}, title = {{Search Problems in Trees with Symmetries: Near Optimal Traversal Strategies for Individualization-Refinement Algorithms}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {16:1--16:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.16}, URN = {urn:nbn:de:0030-drops-140853}, doi = {10.4230/LIPIcs.ICALP.2021.16}, annote = {Keywords: Online algorithms, Graph isomorphism, Lower bounds} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

This paper considers the length of resolution proofs when using Krishnamurthy’s classic symmetry rules. We show that inconsistent linear equation systems of bounded width over a fixed finite field 𝔽_p with p a prime have, in their standard encoding as CNFs, polynomial length resolutions when using the local symmetry rule (SRC-II).
As a consequence it follows that the multipede instances for the graph isomorphism problem encoded as CNF formula have polynomial length resolution proofs. This contrasts exponential lower bounds for individualization-refinement algorithms on these graphs.
For the Cai-Fürer-Immerman graphs, for which Torán showed exponential lower bounds for resolution proofs (SAT 2013), we also show that already the global symmetry rule (SRC-I) suffices to allow for polynomial length proofs.

Pascal Schweitzer and Constantin Seebach. Resolution with Symmetry Rule Applied to Linear Equations. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{schweitzer_et_al:LIPIcs.STACS.2021.58, author = {Schweitzer, Pascal and Seebach, Constantin}, title = {{Resolution with Symmetry Rule Applied to Linear Equations}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {58:1--58:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.58}, URN = {urn:nbn:de:0030-drops-137038}, doi = {10.4230/LIPIcs.STACS.2021.58}, annote = {Keywords: Logical Resolution, Symmetry Rule, Linear Equation System} }

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**Published in:** LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)

In the quest for a logic capturing Ptime the next natural classes of structures to consider are those with bounded color class size. We present a canonization procedure for graphs with dihedral color classes of bounded size in the logic of Choiceless Polynomial Time (CPT), which then captures Ptime on this class of structures. This is the first result of this form for non-abelian color classes.
The first step proposes a normal form which comprises a "rigid assemblage". This roughly means that the local automorphism groups form 2-injective 3-factor subdirect products. Structures with color classes of bounded size can be reduced canonization preservingly to normal form in CPT.
In the second step, we show that for graphs in normal form with dihedral color classes of bounded size, the canonization problem can be solved in CPT. We also show the same statement for general ternary structures in normal form if the dihedral groups are defined over odd domains.

Moritz Lichter and Pascal Schweitzer. Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 31:1-31:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lichter_et_al:LIPIcs.CSL.2021.31, author = {Lichter, Moritz and Schweitzer, Pascal}, title = {{Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time}}, booktitle = {29th EACSL Annual Conference on Computer Science Logic (CSL 2021)}, pages = {31:1--31:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-175-7}, ISSN = {1868-8969}, year = {2021}, volume = {183}, editor = {Baier, Christel and Goubault-Larrecq, Jean}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.31}, URN = {urn:nbn:de:0030-drops-134650}, doi = {10.4230/LIPIcs.CSL.2021.31}, annote = {Keywords: Choiceless polynomial time, canonization, relational structures, bounded color class size, dihedral groups} }

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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-dev.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 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in O(n log n) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For sqrt(n) <= s <= n, we present algorithms that use O(s log n) bits and O(1/s n^2 log n) time for computing the length of a longest increasing subsequence, and O(1/s n^2 log^2 n) time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space.

Masashi Kiyomi, Hirotaka Ono, Yota Otachi, Pascal Schweitzer, and Jun Tarui. Space-Efficient Algorithms for Longest Increasing Subsequence. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kiyomi_et_al:LIPIcs.STACS.2018.44, author = {Kiyomi, Masashi and Ono, Hirotaka and Otachi, Yota and Schweitzer, Pascal and Tarui, Jun}, title = {{Space-Efficient Algorithms for Longest Increasing Subsequence}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.44}, URN = {urn:nbn:de:0030-drops-84911}, doi = {10.4230/LIPIcs.STACS.2018.44}, annote = {Keywords: longest increasing subsequence, patience sorting, space-efficient algorithm} }

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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-dev.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 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

The paper develops a new technique to extract a characteristic subset from a random source that repeatedly samples from a set of elements. Here a characteristic subset is a set that when containing an element contains all elements that have the same probability.
With this technique at hand the paper looks at the special case of the tournament isomorphism problem that stands in the way towards a polynomial-time algorithm for the graph isomorphism problem. Noting that there is a reduction from the automorphism (asymmetry) problem to the isomorphism problem, a reduction in the other direction is nevertheless not known and remains a thorny open problem.
Applying the new technique, we develop a randomized polynomial-time Turing-reduction from the tournament isomorphism problem to the tournament automorphism problem. This is the first such reduction for any kind of combinatorial object not known to have a polynomial-time solvable isomorphism problem.

Pascal Schweitzer. A Polynomial-Time Randomized Reduction from Tournament Isomorphism to Tournament Asymmetry. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{schweitzer:LIPIcs.ICALP.2017.66, author = {Schweitzer, Pascal}, title = {{A Polynomial-Time Randomized Reduction from Tournament Isomorphism to Tournament Asymmetry}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {66:1--66:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.66}, URN = {urn:nbn:de:0030-drops-74928}, doi = {10.4230/LIPIcs.ICALP.2017.66}, annote = {Keywords: graph isomorphism, asymmetry, tournaments, randomized reductions} }

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**Published in:** Dagstuhl Reports, Volume 5, Issue 12 (2016)

This report documents the program and the outcomes of Dagstuhl Seminar 15511 "The Graph Isomorphism Problem". The goal of the seminar was to bring together researchers working on the numerous topics closely related to the
Isomorphism Problem to foster their collaboration. To this end the participants of the seminar included researchers working on the theoretical and practical aspects of isomorphism ranging from the fields of algorithmic group theory, finite model theory, combinatorial optimization to algorithmics.
A highlight of the conference was the presentation of a new quasi-polynomial time algorithm for the Graph Isomorphism Problem, providing the first improvement since 1983.

László Babai, Anuj Dawar, Pascal Schweitzer, and Jacobo Torán. The Graph Isomorphism Problem (Dagstuhl Seminar 15511). In Dagstuhl Reports, Volume 5, Issue 12, pp. 1-17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@Article{babai_et_al:DagRep.5.12.1, author = {Babai, L\'{a}szl\'{o} and Dawar, Anuj and Schweitzer, Pascal and Tor\'{a}n, Jacobo}, title = {{The Graph Isomorphism Problem (Dagstuhl Seminar 15511)}}, pages = {1--17}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2016}, volume = {5}, number = {12}, editor = {Babai, L\'{a}szl\'{o} and Dawar, Anuj and Schweitzer, Pascal and Tor\'{a}n, Jacobo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.5.12.1}, URN = {urn:nbn:de:0030-drops-58028}, doi = {10.4230/DagRep.5.12.1}, annote = {Keywords: canonical forms, complexity, computational group theory, graph isomorphism} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized in deterministic logarithmic space (logspace). This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous classes of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.

Michael Elberfeld and Pascal Schweitzer. Canonizing Graphs of Bounded Tree Width in Logspace. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{elberfeld_et_al:LIPIcs.STACS.2016.32, author = {Elberfeld, Michael and Schweitzer, Pascal}, title = {{Canonizing Graphs of Bounded Tree Width in Logspace}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.32}, URN = {urn:nbn:de:0030-drops-57336}, doi = {10.4230/LIPIcs.STACS.2016.32}, annote = {Keywords: algorithmic graph theory, computational complexity, graph isomorphism, logspace, tree width} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

In this paper we resolve the complexity of the isomorphism problem on
all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color class size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.

Pascal Schweitzer. Towards an Isomorphism Dichotomy for Hereditary Graph Classes. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 689-702, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{schweitzer:LIPIcs.STACS.2015.689, author = {Schweitzer, Pascal}, title = {{Towards an Isomorphism Dichotomy for Hereditary Graph Classes}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {689--702}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.689}, URN = {urn:nbn:de:0030-drops-49513}, doi = {10.4230/LIPIcs.STACS.2015.689}, annote = {Keywords: graph isomorphism, modular decomposition, bounded color valence, reductions, forbidden induced subgraphs} }

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