6 Search Results for "Ritzert, Martin"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.105
NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability

Authors: Prem Nigam Kar, David E. Roberson, Tim Seppelt, and Peter Zeman

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Mančinska and Roberson [FOCS'20] showed that two graphs are quantum isomorphic if and only if they are homomorphism indistinguishable over the class of planar graphs. Atserias et al. [JCTB'19] proved that quantum isomorphism is undecidable in general. The NPA hierarchy gives a sequence of semidefinite programming relaxations of quantum isomorphism. Recently, Roberson and Seppelt [ICALP'23] obtained a homomorphism indistinguishability characterization of the feasibility of each level of the Lasserre hierarchy of semidefinite programming relaxations of graph isomorphism. We prove a quantum analogue of this result by showing that each level of the NPA hierarchy of SDP relaxations for quantum isomorphism of graphs is equivalent to homomorphism indistinguishability over an appropriate class of planar graphs. By combining the convergence of the NPA hierarchy with the fact that the union of these graph classes is the set of all planar graphs, we are able to give a new proof of the result of Mančinska and Roberson [FOCS'20] that avoids the use of the theory of quantum groups. This homomorphism indistinguishability characterization also allows us to give a randomized polynomial-time algorithm deciding exact feasibility of each fixed level of the NPA hierarchy of SDP relaxations for quantum isomorphism.
Cite as

Prem Nigam Kar, David E. Roberson, Tim Seppelt, and Peter Zeman. NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 105:1-105:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kar_et_al:LIPIcs.ICALP.2025.105,
  author =	{Kar, Prem Nigam and Roberson, David E. and Seppelt, Tim and Zeman, Peter},
  title =	{{NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{105:1--105:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.105},
  URN =		{urn:nbn:de:0030-drops-234828},
  doi =		{10.4230/LIPIcs.ICALP.2025.105},
  annote =	{Keywords: Quantum isomorphism, NPA hierarchy, homomorphism indistinguishability}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2025.4
Learning Aggregate Queries Defined by First-Order Logic with Counting

Authors: Steffen van Bergerem and Nicole Schweikardt

Published in: LIPIcs, Volume 328, 28th International Conference on Database Theory (ICDT 2025)

Abstract
In the logical framework introduced by Grohe and Turán (TOCS 2004) for Boolean classification problems, the instances to classify are tuples from a logical structure, and Boolean classifiers are described by parametric models based on logical formulas. This is a specific scenario for supervised passive learning, where classifiers should be learned based on labelled examples. Existing results in this scenario focus on Boolean classification. This paper presents learnability results beyond Boolean classification. We focus on multiclass classification problems where the task is to assign input tuples to arbitrary integers. To represent such integer-valued classifiers, we use aggregate queries specified by an extension of first-order logic with counting terms called FOC₁. Our main result shows the following: given a database of polylogarithmic degree, within quasi-linear time, we can build an index structure that makes it possible to learn FOC₁-definable integer-valued classifiers in time polylogarithmic in the size of the database and polynomial in the number of training examples.
Cite as

Steffen van Bergerem and Nicole Schweikardt. Learning Aggregate Queries Defined by First-Order Logic with Counting. In 28th International Conference on Database Theory (ICDT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 328, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanbergerem_et_al:LIPIcs.ICDT.2025.4,
  author =	{van Bergerem, Steffen and Schweikardt, Nicole},
  title =	{{Learning Aggregate Queries Defined by First-Order Logic with Counting}},
  booktitle =	{28th International Conference on Database Theory (ICDT 2025)},
  pages =	{4:1--4:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-364-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{328},
  editor =	{Roy, Sudeepa and Kara, Ahmet},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2025.4},
  URN =		{urn:nbn:de:0030-drops-229457},
  doi =		{10.4230/LIPIcs.ICDT.2025.4},
  annote =	{Keywords: Supervised learning, multiclass classification problems, counting logic}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.8
The Parameterized Complexity of Learning Monadic Second-Order Logic

Authors: Steffen van Bergerem, Martin Grohe, and Nina Runde

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
Within the model-theoretic framework for supervised learning introduced by Grohe and Turán (TOCS 2004), we study the parameterized complexity of learning concepts definable in monadic second-order logic (MSO). We show that the problem of learning an MSO-definable concept from a training sequence of labeled examples is fixed-parameter tractable on graphs of bounded clique-width, and that it is hard for the parameterized complexity class para-NP on general graphs. It turns out that an important distinction to be made is between 1-dimensional and higher-dimensional concepts, where the instances of a k-dimensional concept are k-tuples of vertices of a graph. For the higher-dimensional case, we give a learning algorithm that is fixed-parameter tractable in the size of the graph, but not in the size of the training sequence, and we give a hardness result showing that this is optimal. By comparison, in the 1-dimensional case, we obtain an algorithm that is fixed-parameter tractable in both.
Cite as

Steffen van Bergerem, Martin Grohe, and Nina Runde. The Parameterized Complexity of Learning Monadic Second-Order Logic. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanbergerem_et_al:LIPIcs.CSL.2025.8,
  author =	{van Bergerem, Steffen and Grohe, Martin and Runde, Nina},
  title =	{{The Parameterized Complexity of Learning Monadic Second-Order Logic}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.8},
  URN =		{urn:nbn:de:0030-drops-227651},
  doi =		{10.4230/LIPIcs.CSL.2025.8},
  annote =	{Keywords: monadic second-order definable concept learning, agnostic probably approximately correct learning, parameterized complexity, clique-width, fixed-parameter tractable, Boolean classification, supervised learning, monadic second-order logic}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.13
Computational Complexity of the Weisfeiler-Leman Dimension

Authors: Moritz Lichter, Simon Raßmann, and Pascal Schweitzer

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
The Weisfeiler-Leman dimension of a graph G is the least number k such that the k-dimensional Weisfeiler-Leman algorithm distinguishes G from every other non-isomorphic graph, or equivalently, the least k such that G is definable in (k+1)-variable first-order logic with counting. The dimension is a standard measure of the descriptive or structural complexity of a graph and recently finds various applications in particular in the context of machine learning. This paper studies the complexity of computing the Weisfeiler-Leman dimension. We observe that deciding whether the Weisfeiler-Leman dimension of G is at most k is NP-hard, even if G is restricted to have 4-bounded color classes. For each fixed k ≥ 2, we give a polynomial-time algorithm that decides whether the Weisfeiler-Leman dimension of a given graph with 5-bounded color classes is at most k. Moreover, we show that for these bounds on the color classes, this is optimal because the problem is PTIME-hard under logspace-uniform AC_0-reductions. Furthermore, for each larger bound c on the color classes and each fixed k ≥ 2, we provide a polynomial-time decision algorithm for the abelian case, that is, for structures of which each color class has an abelian automorphism group. While the graph classes we consider may seem quite restrictive, graphs with 4-bounded abelian colors include CFI-graphs and multipedes, which form the basis of almost all known hard instances and lower bounds related to the Weisfeiler-Leman algorithm.
Cite as

Moritz Lichter, Simon Raßmann, and Pascal Schweitzer. Computational Complexity of the Weisfeiler-Leman Dimension. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lichter_et_al:LIPIcs.CSL.2025.13,
  author =	{Lichter, Moritz and Ra{\ss}mann, Simon and Schweitzer, Pascal},
  title =	{{Computational Complexity of the Weisfeiler-Leman Dimension}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{13:1--13:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.13},
  URN =		{urn:nbn:de:0030-drops-227707},
  doi =		{10.4230/LIPIcs.CSL.2025.13},
  annote =	{Keywords: Weisfeiler-Leman algorithm, dimension, complexity, coherent configurations}
}
Document
DOI: 10.4230/LIPIcs.CSL.2025.15
On the VC Dimension of First-Order Logic with Counting and Weight Aggregation

Authors: Steffen van Bergerem and Nicole Schweikardt

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract
We prove optimal upper bounds on the Vapnik-Chervonenkis density of formulas in the extensions of first-order logic with counting (FOC_1) and with weight aggregation (FOWA_1) on nowhere dense classes of (vertex- and edge-)weighted finite graphs. This lifts a result of Pilipczuk, Siebertz, and Toruńczyk [Michał Pilipczuk et al., 2018] from first-order logic on ordinary finite graphs to substantially more expressive logics on weighted finite graphs. Moreover, this proves that every FOC_1 formula and every FOWA_1 formula has bounded Vapnik-Chervonenkis dimension on nowhere dense classes of weighted finite graphs; thereby, it lifts a result of Adler and Adler [Hans Adler and Isolde Adler, 2014] from first-order logic to FOC_1 and FOWA_1. Generalising another result of Pilipczuk, Siebertz, and Toruńczyk [Michał Pilipczuk et al., 2018], we also provide an explicit upper bound on the ladder index of FOC_1 and FOWA_1 formulas on nowhere dense classes. This shows that nowhere dense classes of weighted finite graphs are FOC_1-stable and FOWA_1-stable.
Cite as

Steffen van Bergerem and Nicole Schweikardt. On the VC Dimension of First-Order Logic with Counting and Weight Aggregation. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanbergerem_et_al:LIPIcs.CSL.2025.15,
  author =	{van Bergerem, Steffen and Schweikardt, Nicole},
  title =	{{On the VC Dimension of First-Order Logic with Counting and Weight Aggregation}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.15},
  URN =		{urn:nbn:de:0030-drops-227722},
  doi =		{10.4230/LIPIcs.CSL.2025.15},
  annote =	{Keywords: VC dimension, VC density, stability, nowhere dense graphs, first-order logic with weight aggregation, first-order logic with counting}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2019.24
Learning Definable Hypotheses on Trees

Authors: Emilie Grienenberger and Martin Ritzert

Published in: LIPIcs, Volume 127, 22nd International Conference on Database Theory (ICDT 2019)

Abstract
We study the problem of learning properties of nodes in tree structures. Those properties are specified by logical formulas, such as formulas from first-order or monadic second-order logic. We think of the tree as a database encoding a large dataset and therefore aim for learning algorithms which depend at most sublinearly on the size of the tree. We present a learning algorithm for quantifier-free formulas where the running time only depends polynomially on the number of training examples, but not on the size of the background structure. By a previous result on strings we know that for general first-order or monadic second-order (MSO) formulas a sublinear running time cannot be achieved. However, we show that by building an index on the tree in a linear time preprocessing phase, we can achieve a learning algorithm for MSO formulas with a logarithmic learning phase.
Cite as

Emilie Grienenberger and Martin Ritzert. Learning Definable Hypotheses on Trees. In 22nd International Conference on Database Theory (ICDT 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 127, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{grienenberger_et_al:LIPIcs.ICDT.2019.24,
  author =	{Grienenberger, Emilie and Ritzert, Martin},
  title =	{{Learning Definable Hypotheses on Trees}},
  booktitle =	{22nd International Conference on Database Theory (ICDT 2019)},
  pages =	{24:1--24:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-101-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{127},
  editor =	{Barcelo, Pablo and Calautti, Marco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2019.24},
  URN =		{urn:nbn:de:0030-drops-103261},
  doi =		{10.4230/LIPIcs.ICDT.2019.24},
  annote =	{Keywords: monadic second-order logic, trees, query learning}
}
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