40 Search Results for "Grohe, Martin"

Document
DOI: 10.4230/LIPIcs.MFCS.2023.82
Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors

Authors: Tim Seppelt

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
Two graphs G and H are homomorphism indistinguishable over a class of graphs ℱ if for all graphs F ∈ ℱ the number of homomorphisms from F to G is equal to the number of homomorphisms from F to H. Many natural equivalence relations comparing graphs such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes. Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics. This result follows from a correspondence between closure properties of a graph class and preservation properties of its homomorphism indistinguishability relation. Furthermore, we classify all graph classes which are in a sense finite (essentially profinite) and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on general properties of the homomorphism distinguishing closure.
Cite as

Tim Seppelt. Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 82:1-82:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{seppelt:LIPIcs.MFCS.2023.82,
  author =	{Seppelt, Tim},
  title =	{{Logical Equivalences, Homomorphism Indistinguishability, and Forbidden Minors}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{82:1--82:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.82},
  URN =		{urn:nbn:de:0030-drops-186161},
  doi =		{10.4230/LIPIcs.MFCS.2023.82},
  annote =	{Keywords: homomorphism indistinguishability, graph minor, logic}
}
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.SEA.2023.11
CompDP: A Framework for Simultaneous Subgraph Counting Under Connectivity Constraints

Authors: Kengo Nakamura, Masaaki Nishino, Norihito Yasuda, and Shin-ichi Minato

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

Abstract
The subgraph counting problem computes the number of subgraphs of a given graph that satisfy some constraints. Among various constraints imposed on a graph, those regarding the connectivity of vertices, such as "these two vertices must be connected," have great importance since they are indispensable for determining various graph substructures, e.g., paths, Steiner trees, and rooted spanning forests. In this view, the subgraph counting problem under connectivity constraints is also important because counting such substructures often corresponds to measuring the importance of a vertex in network infrastructures. However, we must solve the subgraph counting problems multiple times to compute such an importance measure for every vertex. Conventionally, they are solved separately by constructing decision diagrams such as BDD and ZDD for each problem. However, even solving a single subgraph counting is a computationally hard task, preventing us from solving it multiple times in a reasonable time. In this paper, we propose a dynamic programming framework that simultaneously counts subgraphs for every vertex by focusing on similar connectivity constraints. Experimental results show that the proposed method solved multiple subgraph counting problems about 10-20 times faster than the existing approach for many problem settings.
Cite as

Kengo Nakamura, Masaaki Nishino, Norihito Yasuda, and Shin-ichi Minato. CompDP: A Framework for Simultaneous Subgraph Counting Under Connectivity Constraints. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 11:1-11:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{nakamura_et_al:LIPIcs.SEA.2023.11,
  author =	{Nakamura, Kengo and Nishino, Masaaki and Yasuda, Norihito and Minato, Shin-ichi},
  title =	{{CompDP: A Framework for Simultaneous Subgraph Counting Under Connectivity Constraints}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{11:1--11:20},
  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.2023.11},
  URN =		{urn:nbn:de:0030-drops-183613},
  doi =		{10.4230/LIPIcs.SEA.2023.11},
  annote =	{Keywords: Subgraph counting, Connectivity, Zero-suppressed Binary Decision Diagram}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2023.20
Probabilistic Query Evaluation with Bag Semantics

Authors: Martin Grohe, Peter Lindner, and Christoph Standke

Published in: LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

Abstract
We initiate the study of probabilistic query evaluation under bag semantics where tuples are allowed to be present with duplicates. We focus on self-join free conjunctive queries, and probabilistic databases where occurrences of different facts are independent, which is the natural generalization of tuple-independent probabilistic databases to the bag semantics setting. For set semantics, the data complexity of this problem is well understood, even for the more general class of unions of conjunctive queries: it is either in polynomial time, or #P-hard, depending on the query (Dalvi & Suciu, JACM 2012). Due to potentially unbounded multiplicities, the bag probabilistic databases we discuss are no longer finite objects, which requires a treatment of representation mechanisms. Moreover, the answer to a Boolean query is a probability distribution over non-negative integers, rather than a probability distribution over {true, false}. Therefore, we discuss two flavors of probabilistic query evaluation: computing expectations of answer tuple multiplicities, and computing the probability that a tuple is contained in the answer at most k times for some parameter k. Subject to mild technical assumptions on the representation systems, it turns out that expectations are easy to compute, even for unions of conjunctive queries. For query answer probabilities, we obtain a dichotomy between solvability in polynomial time and #P-hardness for self-join free conjunctive queries.
Cite as

Martin Grohe, Peter Lindner, and Christoph Standke. Probabilistic Query Evaluation with Bag Semantics. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 20:1-20:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{grohe_et_al:LIPIcs.ICDT.2023.20,
  author =	{Grohe, Martin and Lindner, Peter and Standke, Christoph},
  title =	{{Probabilistic Query Evaluation with Bag Semantics}},
  booktitle =	{26th International Conference on Database Theory (ICDT 2023)},
  pages =	{20:1--20:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-270-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{255},
  editor =	{Geerts, Floris and Vandevoort, Brecht},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2023.20},
  URN =		{urn:nbn:de:0030-drops-177636},
  doi =		{10.4230/LIPIcs.ICDT.2023.20},
  annote =	{Keywords: Probabilistic Query Evaluation, Probabilistic Databases, Bag Semantics}
}
Document
DOI: 10.4230/DagRep.12.5.112
The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201)

Authors: Martin Grohe, Venkatesan Guruswami, Dániel Marx, and Stanislav Živný

Published in: Dagstuhl Reports, Volume 12, Issue 5 (2022)

Abstract
Constraint satisfaction has always played a central role in computational complexity theory; appropriate versions of CSPs are classical complete problems for most standard complexity classes. CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena. For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non-trivially approximable, fixed-parameter tractable, or definable in a weak logic). In the last 15 years, research activity in this area has significantly intensified and hugely impressive progress was made. The Dagstuhl Seminar 22201 "The Constraint Satisfaction Problem: Complexity and Approximability" was aimed at bringing together researchers using all the different techniques in the study of the CSP so that they can share their insights obtained during the past four years. This report documents the material presented during the course of the seminar.
Cite as

Martin Grohe, Venkatesan Guruswami, Dániel Marx, and Stanislav Živný. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201). In Dagstuhl Reports, Volume 12, Issue 5, pp. 112-130, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Article{grohe_et_al:DagRep.12.5.112,
  author =	{Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav},
  title =	{{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201)}},
  pages =	{112--130},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2022},
  volume =	{12},
  number =	{5},
  editor =	{Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.5.112},
  URN =		{urn:nbn:de:0030-drops-174453},
  doi =		{10.4230/DagRep.12.5.112},
  annote =	{Keywords: Constraint satisfaction problem (CSP); Computational complexity; Hardness of approximation; Universal algebra; Semidefinite programming}
}
Document
DOI: 10.4230/DagRep.12.3.141
Graph Embeddings: Theory meets Practice (Dagstuhl Seminar 22132)

Authors: Martin Grohe, Stephan Günnemann, Stefanie Jegelka, and Christopher Morris

Published in: Dagstuhl Reports, Volume 12, Issue 3 (2022)

Abstract
Vectorial representations of graphs and relational structures, so-called graph embeddings, make it possible to apply standard tools from data mining, machine learning, and statistics to the graph domain. In particular, graph embeddings aim to capture important information about, both, the graph structure and available side information as a vector, to enable downstream tasks such as classification, regression, or clustering. Starting from the 1960s in chemoinformatics, research in various communities has resulted in a plethora of approaches, often with recurring ideas. However, most of the field advancements are driven by intuition and empiricism, often tailored to a specific application domain. Until recently, the area has received little stimulus from theoretical computer science, graph theory, and learning theory. The Dagstuhl Seminar 22132 "Graph Embeddings: Theory meets Practice", was aimed to gather leading applied and theoretical researchers in graph embeddings and adjacent areas, such as graph isomorphism, bio- and chemoinformatics, and graph theory, to stimulate an increased exchange of ideas between these communities.
Cite as

Martin Grohe, Stephan Günnemann, Stefanie Jegelka, and Christopher Morris. Graph Embeddings: Theory meets Practice (Dagstuhl Seminar 22132). In Dagstuhl Reports, Volume 12, Issue 3, pp. 141-155, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Article{grohe_et_al:DagRep.12.3.141,
  author =	{Grohe, Martin and G\"{u}nnemann, Stephan and Jegelka, Stefanie and Morris, Christopher},
  title =	{{Graph Embeddings: Theory meets Practice (Dagstuhl Seminar 22132)}},
  pages =	{141--155},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2022},
  volume =	{12},
  number =	{3},
  editor =	{Grohe, Martin and G\"{u}nnemann, Stephan and Jegelka, Stefanie and Morris, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.3.141},
  URN =		{urn:nbn:de:0030-drops-172727},
  doi =		{10.4230/DagRep.12.3.141},
  annote =	{Keywords: Machine Learning For Graphs, GNNs, Graph Embedding}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.52
Graph Similarity Based on Matrix Norms

Authors: Timo Gervens and Martin Grohe

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
Quantifying the similarity between two graphs is a fundamental algorithmic problem at the heart of many data analysis tasks for graph-based data. In this paper, we study the computational complexity of a family of similarity measures based on quantifying the mismatch between the two graphs, that is, the "symmetric difference" of the graphs under an optimal alignment of the vertices. An important example is similarity based on graph edit distance. While edit distance calculates the "global" mismatch, that is, the number of edges in the symmetric difference, our main focus is on "local" measures calculating the maximum mismatch per vertex. Mathematically, our similarity measures are best expressed in terms of the adjacency matrices: the mismatch between graphs is expressed as the difference of their adjacency matrices (under an optimal alignment), and we measure it by applying some matrix norm. Roughly speaking, global measures like graph edit distance correspond to entrywise matrix norms like the Frobenius norm and local measures correspond to operator norms like the spectral norm. We prove a number of strong NP-hardness and inapproximability results even for very restricted graph classes such as bounded-degree trees.
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Timo Gervens and Martin Grohe. Graph Similarity Based on Matrix Norms. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 52:1-52:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gervens_et_al:LIPIcs.MFCS.2022.52,
  author =	{Gervens, Timo and Grohe, Martin},
  title =	{{Graph Similarity Based on Matrix Norms}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{52:1--52:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.52},
  URN =		{urn:nbn:de:0030-drops-168509},
  doi =		{10.4230/LIPIcs.MFCS.2022.52},
  annote =	{Keywords: graph similarity, approximate graph isomorphism, graph matching}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.70
Homomorphism Tensors and Linear Equations

Authors: Martin Grohe, Gaurav Rattan, and Tim Seppelt

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

Abstract
Lovász (1967) showed that two graphs G and H are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph F, the number of homomorphisms from F to G equals the number of homomorphisms from F to H. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over several natural graph classes, namely trees of bounded degree, graphs of bounded pathwidth (answering a question of Dell et al. (2018)), and graphs of bounded treedepth.
Cite as

Martin Grohe, Gaurav Rattan, and Tim Seppelt. Homomorphism Tensors and Linear Equations. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 70:1-70:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{grohe_et_al:LIPIcs.ICALP.2022.70,
  author =	{Grohe, Martin and Rattan, Gaurav and Seppelt, Tim},
  title =	{{Homomorphism Tensors and Linear Equations}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{70:1--70:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.70},
  URN =		{urn:nbn:de:0030-drops-164113},
  doi =		{10.4230/LIPIcs.ICALP.2022.70},
  annote =	{Keywords: homomorphisms, labelled graphs, treewidth, pathwidth, treedepth, linear equations, Sherali-Adams relaxation, Wiegmann-Specht Theorem, Weisfeiler-Leman}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.MFCS.2021.2
A Deep Dive into the Weisfeiler-Leman Algorithm (Invited Talk)

Authors: Martin Grohe

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
The Weisfeiler-Leman algorithm is a well-known combinatorial graph isomorphism test going back to work of Weisfeiler and Leman in the late 1960s. The algorithm has a surprising number of seemingly unrelated characterisations in terms of logic, algebra, linear and semi-definite programming, and graph homomorphisms. Due to its simplicity and efficiency, it is an important subroutine of all modern graph isomorphism tools. In recent years, further applications in linear optimisation, probabilistic inference, and machine learning have surfaced. In my talk, I will introduce the Weisfeiler-Leman algorithm and some extensions. I will discuss its expressiveness and the various characterisations, and I will speak about its applications.
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Martin Grohe. A Deep Dive into the Weisfeiler-Leman Algorithm (Invited Talk). In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, p. 2:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grohe:LIPIcs.MFCS.2021.2,
  author =	{Grohe, Martin},
  title =	{{A Deep Dive into the Weisfeiler-Leman Algorithm}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.2},
  URN =		{urn:nbn:de:0030-drops-144429},
  doi =		{10.4230/LIPIcs.MFCS.2021.2},
  annote =	{Keywords: Weisfeiler-Leman algorithm, graph isomorphism, counting homomorphisms, finite variable logics}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.37
On the Relative Power of Linear Algebraic Approximations of Graph Isomorphism

Authors: Anuj Dawar and Danny Vagnozzi

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
We compare the capabilities of two approaches to approximating graph isomorphism using linear algebraic methods: the invertible map tests (introduced by Dawar and Holm) and proof systems with algebraic rules, namely polynomial calculus, monomial calculus and Nullstellensatz calculus. In the case of fields of characteristic zero, these variants are all essentially equivalent to the Weisfeiler-Leman algorithms. In positive characteristic we show that the distinguishing power of the monomial calculus is no greater than the invertible map method by simulating the former in a fixed-point logic with solvability operators. In turn, we show that the distinctions made by this logic can be implemented in the Nullstellensatz calculus.
Cite as

Anuj Dawar and Danny Vagnozzi. On the Relative Power of Linear Algebraic Approximations of Graph Isomorphism. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 37:1-37:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dawar_et_al:LIPIcs.MFCS.2021.37,
  author =	{Dawar, Anuj and Vagnozzi, Danny},
  title =	{{On the Relative Power of Linear Algebraic Approximations of Graph Isomorphism}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{37:1--37:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.37},
  URN =		{urn:nbn:de:0030-drops-144774},
  doi =		{10.4230/LIPIcs.MFCS.2021.37},
  annote =	{Keywords: Graph isomorphism, proof complexity, invertible map tests}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2021.134
Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

Authors: Martin Grohe and Sandra Kiefer

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

Abstract
The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm. We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs. The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth.
Cite as

Martin Grohe and Sandra Kiefer. Logarithmic Weisfeiler-Leman Identifies All Planar Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 134:1-134:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grohe_et_al:LIPIcs.ICALP.2021.134,
  author =	{Grohe, Martin and Kiefer, Sandra},
  title =	{{Logarithmic Weisfeiler-Leman Identifies All Planar Graphs}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{134:1--134:20},
  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.2021.134},
  URN =		{urn:nbn:de:0030-drops-142035},
  doi =		{10.4230/LIPIcs.ICALP.2021.134},
  annote =	{Keywords: Weisfeiler-Leman algorithm, finite-variable logic, isomorphism testing, planar graphs, quantifier depth, iteration number}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2021.16
Database Repairing with Soft Functional Dependencies

Authors: Nofar Carmeli, Martin Grohe, Benny Kimelfeld, Ester Livshits, and Muhammad Tibi

Published in: LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)

Abstract
A common interpretation of soft constraints penalizes the database for every violation of every constraint, where the penalty is the cost (weight) of the constraint. A computational challenge is that of finding an optimal subset: a collection of database tuples that minimizes the total penalty when each tuple has a cost of being excluded. When the constraints are strict (i.e., have an infinite cost), this subset is a "cardinality repair" of an inconsistent database; in soft interpretations, this subset corresponds to a "most probable world" of a probabilistic database, a "most likely intention" of a probabilistic unclean database, and so on. Within the class of functional dependencies, the complexity of finding a cardinality repair is thoroughly understood. Yet, very little is known about the complexity of finding an optimal subset for the more general soft semantics. This paper makes a significant progress in this direction. In addition to general insights about the hardness and approximability of the problem, we present algorithms for two special cases: a single functional dependency, and a bipartite matching. The latter is the problem of finding an optimal "almost matching" of a bipartite graph where a penalty is paid for every lost edge and every violation of monogamy.
Cite as

Nofar Carmeli, Martin Grohe, Benny Kimelfeld, Ester Livshits, and Muhammad Tibi. Database Repairing with Soft Functional Dependencies. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 16:1-16:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{carmeli_et_al:LIPIcs.ICDT.2021.16,
  author =	{Carmeli, Nofar and Grohe, Martin and Kimelfeld, Benny and Livshits, Ester and Tibi, Muhammad},
  title =	{{Database Repairing with Soft Functional Dependencies}},
  booktitle =	{24th International Conference on Database Theory (ICDT 2021)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-179-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{186},
  editor =	{Yi, Ke and Wei, Zhewei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.16},
  URN =		{urn:nbn:de:0030-drops-137245},
  doi =		{10.4230/LIPIcs.ICDT.2021.16},
  annote =	{Keywords: Database inconsistency, database repairs, integrity constraints, soft constraints, functional dependencies}
}
Document
DOI: 10.4230/LIPIcs.CSL.2021.10
Learning Concepts Described By Weight Aggregation Logic

Authors: Steffen van Bergerem and Nicole Schweikardt

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

Abstract
We consider weighted structures, which extend ordinary relational structures by assigning weights, i.e. elements from a particular group or ring, to tuples present in the structure. We introduce an extension of first-order logic that allows to aggregate weights of tuples, compare such aggregates, and use them to build more complex formulas. We provide locality properties of fragments of this logic including Feferman-Vaught decompositions and a Gaifman normal form for a fragment called FOW₁, as well as a localisation theorem for a larger fragment called FOWA₁. This fragment can express concepts from various machine learning scenarios. Using the locality properties, we show that concepts definable in FOWA₁ over a weighted background structure of at most polylogarithmic degree are agnostically PAC-learnable in polylogarithmic time after pseudo-linear time preprocessing.
Cite as

Steffen van Bergerem and Nicole Schweikardt. Learning Concepts Described By Weight Aggregation Logic. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 10:1-10:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{vanbergerem_et_al:LIPIcs.CSL.2021.10,
  author =	{van Bergerem, Steffen and Schweikardt, Nicole},
  title =	{{Learning Concepts Described By Weight Aggregation Logic}},
  booktitle =	{29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
  pages =	{10:1--10:18},
  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.2021.10},
  URN =		{urn:nbn:de:0030-drops-134447},
  doi =		{10.4230/LIPIcs.CSL.2021.10},
  annote =	{Keywords: first-order definable concept learning, agnostic probably approximately correct learning, classification problems, locality, Feferman-Vaught decomposition, Gaifman normal form, first-order logic with counting, weight aggregation logic}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.66
A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights

Authors: Artem Govorov, Jin-Yi Cai, and Martin Dyer

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

Abstract
We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix A. Each symmetric matrix A defines a graph homomorphism function Z_A(⋅), also known as the partition function. Dyer and Greenhill [Martin E. Dyer and Catherine S. Greenhill, 2000] established a complexity dichotomy of Z_A(⋅) for symmetric {0, 1}-matrices A, and they further proved that its #P-hardness part also holds for bounded degree graphs. Bulatov and Grohe [Andrei Bulatov and Martin Grohe, 2005] extended the Dyer-Greenhill dichotomy to nonnegative symmetric matrices A. However, their hardness proof requires graphs of arbitrarily large degree, and whether the bounded degree part of the Dyer-Greenhill dichotomy can be extended has been an open problem for 15 years. We resolve this open problem and prove that for nonnegative symmetric A, either Z_A(G) is in polynomial time for all graphs G, or it is #P-hard for bounded degree (and simple) graphs G. We further extend the complexity dichotomy to include nonnegative vertex weights. Additionally, we prove that the #P-hardness part of the dichotomy by Goldberg et al. [Leslie A. Goldberg et al., 2010] for Z_A(⋅) also holds for simple graphs, where A is any real symmetric matrix.
Cite as

Artem Govorov, Jin-Yi Cai, and Martin Dyer. A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 66:1-66:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{govorov_et_al:LIPIcs.ICALP.2020.66,
  author =	{Govorov, Artem and Cai, Jin-Yi and Dyer, Martin},
  title =	{{A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{66:1--66:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.66},
  URN =		{urn:nbn:de:0030-drops-124733},
  doi =		{10.4230/LIPIcs.ICALP.2020.66},
  annote =	{Keywords: Graph homomorphism, Complexity dichotomy, Counting problems}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2020.16
Infinite Probabilistic Databases

Authors: Martin Grohe and Peter Lindner

Published in: LIPIcs, Volume 155, 23rd International Conference on Database Theory (ICDT 2020)

Abstract
Probabilistic databases (PDBs) are used to model uncertainty in data in a quantitative way. In the standard formal framework, PDBs are finite probability spaces over relational database instances. It has been argued convincingly that this is not compatible with an open-world semantics (Ceylan et al., KR 2016) and with application scenarios that are modeled by continuous probability distributions (Dalvi et al., CACM 2009). We recently introduced a model of PDBs as infinite probability spaces that addresses these issues (Grohe and Lindner, PODS 2019). While that work was mainly concerned with countably infinite probability spaces, our focus here is on uncountable spaces. Such an extension is necessary to model typical continuous probability distributions that appear in many applications. However, an extension beyond countable probability spaces raises nontrivial foundational issues concerned with the measurability of events and queries and ultimately with the question whether queries have a well-defined semantics. It turns out that so-called finite point processes are the appropriate model from probability theory for dealing with probabilistic databases. This model allows us to construct suitable (uncountable) probability spaces of database instances in a systematic way. Our main technical results are measurability statements for relational algebra queries as well as aggregate queries and Datalog queries.
Cite as

Martin Grohe and Peter Lindner. Infinite Probabilistic Databases. In 23rd International Conference on Database Theory (ICDT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 155, pp. 16:1-16:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{grohe_et_al:LIPIcs.ICDT.2020.16,
  author =	{Grohe, Martin and Lindner, Peter},
  title =	{{Infinite Probabilistic Databases}},
  booktitle =	{23rd International Conference on Database Theory (ICDT 2020)},
  pages =	{16:1--16:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-139-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{155},
  editor =	{Lutz, Carsten and Jung, Jean Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2020.16},
  URN =		{urn:nbn:de:0030-drops-119400},
  doi =		{10.4230/LIPIcs.ICDT.2020.16},
  annote =	{Keywords: Probabilistic Databases, Possible Worlds Semantics, Query Measurability, Relational Algebra, Aggregate Queries}
}
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