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DOI: 10.4230/LIPIcs.ESA.2023.11

Exploration of Graphs with Excluded Minors

Authors: Júlia Baligács, Yann Disser, Irene Heinrich, and Pascal Schweitzer

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

Abstract

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).

Cite as

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.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}
}

Document

DOI: 10.4230/LIPIcs.SAT.2023.1

Algorithms Transcending the SAT-Symmetry Interface

Authors: Markus Anders, Pascal Schweitzer, and Mate Soos

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

Abstract

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.

Cite as

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.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}
}

Document

DOI: 10.4230/LIPIcs.SEA.2023.1

Engineering a Preprocessor for Symmetry Detection

Authors: Markus Anders, Pascal Schweitzer, and Julian Stieß

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

Abstract

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.

Cite as

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}
}

Document

DOI: 10.4230/LIPIcs.ESA.2022.27

A Systematic Study of Isomorphism Invariants of Finite Groups via the Weisfeiler-Leman Dimension

Authors: Jendrik Brachter and Pascal Schweitzer

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

Abstract

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.

Cite as

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.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}
}

Document

DOI: 10.4230/LIPIcs.SAT.2022.1

SAT Preprocessors and Symmetry

Authors: Markus Anders

Published in: LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)

Abstract

Exploitation of symmetries is an indispensable approach to solve certain classes of difficult SAT instances. Numerous techniques for the use of symmetry in SAT have evolved over the past few decades. But no matter how symmetries are used precisely, they have to be detected first. We investigate how to detect more symmetry, faster. The initial idea is to reap the benefits of SAT preprocessing for symmetry detection. As it turns out, applying an off-the-shelf preprocessor before handling symmetry runs into problems: the preprocessor can haphazardly remove symmetry from formulas, severely impeding symmetry exploitation. Our main contribution is a theoretical framework that captures the relationship of SAT preprocessing techniques and symmetry. Based on this, we create a symmetry-aware preprocessor that can be applied safely before handling symmetry. We then demonstrate that applying the preprocessor does not only substantially decrease symmetry detection and breaking times, but also uncovers hidden symmetry not detectable in the original instances. Overall, we depart the conventional view of treating symmetry detection as a black-box, presenting a new application-specific approach to symmetry detection in SAT.

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Markus Anders. SAT Preprocessors and Symmetry. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{anders:LIPIcs.SAT.2022.1,
  author =	{Anders, Markus},
  title =	{{SAT Preprocessors and Symmetry}},
  booktitle =	{25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-242-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{236},
  editor =	{Meel, Kuldeep S. and Strichman, Ofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.1},
  URN =		{urn:nbn:de:0030-drops-166752},
  doi =		{10.4230/LIPIcs.SAT.2022.1},
  annote =	{Keywords: boolean satisfiability, symmetry exploitation, graph isomorphism}
}

Document

DOI: 10.4230/LIPIcs.CSL.2022.36

Number of Variables for Graph Differentiation and the Resolution of GI Formulas

Authors: Jacobo Torán and Florian Wörz

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract

We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and positive depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution. Using this connection, we obtain upper and lower bounds for refuting graph isomorphism formulas in (normal) resolution. In particular, we show that if k is the number of variables needed to distinguish two graphs with n vertices each, then there is an n^O(k) resolution refutation size upper bound for the corresponding isomorphism formula, as well as lower bounds of 2^(k-1) and k for the tree-like resolution size and resolution clause space for this formula. We also show a (normal) resolution size lower bound of exp(Ω(k²/n)) for the case of colored graphs with constant color class sizes. Applying these results, we prove the first exponential lower bound for graph isomorphism formulas in the proof system SRC-1, a system that extends resolution with a global symmetry rule, thereby answering an open question posed by Schweitzer and Seebach.

Cite as

Jacobo Torán and Florian Wörz. Number of Variables for Graph Differentiation and the Resolution of GI Formulas. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{toran_et_al:LIPIcs.CSL.2022.36,
  author =	{Tor\'{a}n, Jacobo and W\"{o}rz, Florian},
  title =	{{Number of Variables for Graph Differentiation and the Resolution of GI Formulas}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.36},
  URN =		{urn:nbn:de:0030-drops-157564},
  doi =		{10.4230/LIPIcs.CSL.2022.36},
  annote =	{Keywords: Proof Complexity, Resolution, Narrow Width, Graph Isomorphism, k-variable fragment first-order logic 𝔏\underlinek, Immerman’s Pebble Game, Symmetry Rule, SRC-1}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2021.24

A Characterization of Individualization-Refinement Trees

Authors: Markus Anders, Jendrik Brachter, and Pascal Schweitzer

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

Abstract

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.

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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.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

DOI: 10.4230/LIPIcs.ESA.2021.6

Parallel Computation of Combinatorial Symmetries

Authors: Markus Anders and Pascal Schweitzer

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

Abstract

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.

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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.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}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2021.15

Comparative Design-Choice Analysis of Color Refinement Algorithms Beyond the Worst Case

Authors: Markus Anders, Pascal Schweitzer, and Florian Wetzels

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

Abstract

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.

Cite as

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.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

DOI: 10.4230/LIPIcs.ICALP.2021.16

Search Problems in Trees with Symmetries: Near Optimal Traversal Strategies for Individualization-Refinement Algorithms

Authors: Markus Anders and Pascal Schweitzer

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

Abstract

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.

Cite as

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.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}
}

Document

DOI: 10.4230/LIPIcs.STACS.2021.58

Resolution with Symmetry Rule Applied to Linear Equations

Authors: Pascal Schweitzer and Constantin Seebach

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

Abstract

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.

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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.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}
}

Document

DOI: 10.4230/LIPIcs.CSL.2021.31

Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time

Authors: Moritz Lichter and Pascal Schweitzer

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

Abstract

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.

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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.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}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2020.4

Isomorphism Problem for S_d-Graphs

Authors: Deniz Ağaoğlu and Petr Hliněný

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract

An H-graph is the intersection graph of connected subgraphs of a suitable subdivision of a fixed graph H, introduced by Biró, Hujter and Tuza (1992). We focus on S_d-graphs as a special case generalizing interval graphs. A graph G is an S_d-graph iff it is the intersection graph of connected subgraphs of a subdivision of a star S_d with d rays. We give an FPT algorithm to solve the isomorphism problem for S_d-graphs with the parameter d. This solves an open problem of Chaplick, Töpfer, Voborník and Zeman (2016). In the course of our proof, we also show that the isomorphism problem of S_d-graphs is computationally at least as hard as the isomorphism problem of posets of bounded width.

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Deniz Ağaoğlu and Petr Hliněný. Isomorphism Problem for S_d-Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{agaoglu_et_al:LIPIcs.MFCS.2020.4,
  author =	{A\u{g}ao\u{g}lu, Deniz and Hlin\v{e}n\'{y}, Petr},
  title =	{{Isomorphism Problem for S\underlined-Graphs}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.4},
  URN =		{urn:nbn:de:0030-drops-126754},
  doi =		{10.4230/LIPIcs.MFCS.2020.4},
  annote =	{Keywords: intersection graph, isomorphism testing, interval graph, H-graph}
}

Document

DOI: 10.4230/LIPIcs.STACS.2020.43

Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm

Authors: Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract

It is well known that the isomorphism problem for vertex-colored graphs with color multiplicity at most 3 is solvable by the classical 2-dimensional Weisfeiler-Leman algorithm (2-WL). On the other hand, the prominent Cai-Fürer-Immerman construction shows that even the multidimensional version of the algorithm does not suffice for graphs with color multiplicity 4. We give an efficient decision procedure that, given a graph G of color multiplicity 4, recognizes whether or not G is identifiable by 2-WL, that is, whether or not 2-WL distinguishes G from any non-isomorphic graph. In fact, we solve the more general problem of recognizing whether or not a given coherent configuration of maximum fiber size 4 is separable. This extends our recognition algorithm to directed graphs of color multiplicity 4 with colored edges. Our decision procedure is based on an explicit description of the class of graphs with color multiplicity 4 that are not identifiable by 2-WL. The Cai-Fürer-Immerman graphs of color multiplicity 4 distinctly appear here as a natural subclass, which demonstrates that the Cai-Fürer-Immerman construction is not ad hoc. Our classification reveals also other types of graphs that are hard for 2-WL. One of them arises from patterns known as (n₃)-configurations in incidence geometry.

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Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fuhlbruck_et_al:LIPIcs.STACS.2020.43,
  author =	{Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg},
  title =	{{Identifiability of Graphs with Small Color Classes by the Weisfeiler-Leman Algorithm}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{43:1--43:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.43},
  URN =		{urn:nbn:de:0030-drops-119046},
  doi =		{10.4230/LIPIcs.STACS.2020.43},
  annote =	{Keywords: Graph Isomorphism, Weisfeiler-Leman Algorithm, Cai-F\"{u}rer-Immerman Graphs, coherent Configurations}
}

Document

DOI: 10.4230/LIPIcs.STACS.2019.7

Best-Of-Two-Worlds Analysis of Online Search

Authors: Spyros Angelopoulos, Christoph Dürr, and Shendan Jin

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract

In search problems, a mobile searcher seeks to locate a target that hides in some unknown position of the environment. Such problems are typically considered to be of an on-line nature, in that the input is unknown to the searcher, and the performance of a search strategy is usually analyzed by means of the standard framework of the competitive ratio, which compares the cost incurred by the searcher to an optimal strategy that knows the location of the target. However, one can argue that even for simple search problems, competitive analysis fails to distinguish between strategies which, intuitively, should have different performance in practice. Motivated by the above, in this work we introduce and study measures supplementary to competitive analysis in the context of search problems. In particular, we focus on the well-known problem of linear search, informally known as the cow-path problem, for which there is an infinite number of strategies that achieve an optimal competitive ratio equal to 9. We propose a measure that reflects the rate at which the line is being explored by the searcher, and which can be seen as an extension of the bijective ratio over an uncountable set of requests. Using this measure we show that a natural strategy that explores the line aggressively is optimal among all 9-competitive strategies. This provides, in particular, a strict separation from the competitively optimal doubling strategy, which is much more conservative in terms of exploration. We also provide evidence that this aggressiveness is requisite for optimality, by showing that any optimal strategy must mimic the aggressive strategy in its first few explorations.

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Spyros Angelopoulos, Christoph Dürr, and Shendan Jin. Best-Of-Two-Worlds Analysis of Online Search. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{angelopoulos_et_al:LIPIcs.STACS.2019.7,
  author =	{Angelopoulos, Spyros and D\"{u}rr, Christoph and Jin, Shendan},
  title =	{{Best-Of-Two-Worlds Analysis of Online Search}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.7},
  URN =		{urn:nbn:de:0030-drops-102467},
  doi =		{10.4230/LIPIcs.STACS.2019.7},
  annote =	{Keywords: Online computation, search problems, linear search, performance measures}
}

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