Talks (Archive)

Showing 1 - 50 of 173 Results

  • Date
     
    Title
  • 12/11/2020
    11:00 - 11:30
     
    Connectivity and Routing using Graph Neural Networks
    Frorian Frantzen
    Graph Neural Networks form a deep learning architecture for machine learning tasks on graphs. Recently, they have been used to obtain heuristics for hard combinatorial problems such as TSP, SAT, or constraint satisfaction problems. The purpose of this thesis is to investigate how such approaches perform on computationally much easier shortest path problems. We will present lower bounds regarding the depth and width of a message passing neural network, which must be fulfilled for such a model to learn the shortest path problem. Then we will evaluate an existing neural network -- originally designed for the traveling salesperson problem -- for its applicability to the shortest path problem. While this network will fail with insignificant accuracies, these tests will allow us to derive unique challenges of the shortest path problem and which weaknesses in the architecture we need to fix to achieve better results. Based on these results, we will built our own neural network architecture and train it both supervised and unsupervised. We will evaluate this network in terms accuracy, scalability to larger graphs and how well the network generalizes the shortest path problem.
  • 30/10/2020
    10:00 - 11:00
    ATI
    Flag Algebras in Graph Theory
    Tim Seppelt
    Flag algebras as introduced by Razborov represent a powerful tool in extremal combinatorics. This talk will provide a gentle introduction to the framework highlighting applications graph theory. Moreover, limitations of flag algebraic techniques are discussed following the work of Hatami and Norine.
  • 30/10/2020
    10:00 - 11:00
    ATI
    Flag Algebras in Graph Theory
    Tim Seppelt
    Flag algebras as introduced by Razborov represent a powerful tool in extremal combinatorics. This talk will provide a gentle introduction to the framework highlighting applications graph theory. Moreover, limitations of flag algebraic techniques are discussed following the work of Hatami and Norine.
  • 27/10/2020
    14:00 - 14:30
    BSc
    The Fine-Grained Complexity of Longest Common Subsequence
    Benjamin Stutte
    [bachelor]We will examine approaches to fine-grained analyses of the Longest Common Subsequence (LCS) problem with special attention to recent results that have proven lower bounds for solving LCS under the Strong Exponential Time Hypothesis. We will first showcase two results, one by Abboud et al. (FOCS’15) and the other by Bringmann et al. (FOCS’15), where sub-quadratic lower bounds have been proven for the general LCS problem. Later, we turn to Bringmann et al.’s (SODA’18) which examined the optimal running time of multivariate LCS which proved lower bounds that coincide with the running times of the fastest known algorithms.
  • 27/10/2020
    14:45 - 15:15
    BSc
    Stable and Efficient Algorithms for Logarithmically Counting Homomorphisms from Selected Graph Classes
    Anton Florey
    Graph homomorphisms are mappings between two graphs H and G that preserve all edge relationsof the left-hand side graph. The graph homomorphism counting problem #Hom(H, G) asks for
    the number of homomorphisms from one graph H ∈ H to another graph G ∈ G. It is polynomial
    time solvable for certain left-hand graph classes H. As part of his recent master thesis, Maximilian
    Merz already provided implementations of various homomorphism counting algorithms. Unfor-
    tunately, they quickly get impracticable for large problem instances, as homomorphism numbers
    grow extremely fast. This work revisits most of these algorithms with the novel idea of counting
    the logarithm of these numbers. One main contribution is the implementation and evaluation of
    these adapted algorithms. For most of the newly implemented functions, a significant improve-
    ment in efficiency over the exact counting implementations can be measured. Further experiments
    also show that errors of this logarithmic approximation stay sufficiently close to the unavoidable

    accuracy loss induced by finite machine precision.
    https://rwth.zoom.us/j/95988725888?pwd=NUlaM05VdU5Kdkp0NXVTZHplWlFSQT09
  • 27/10/2020
    14:00 - 14:30
    BSc
    The Fine-Grained Complexity of Longest Common Subsequence
    Benjamin Stutte
    [bachelor]We will examine approaches to fine-grained analyses of the Longest Common Subsequence (LCS) problem with special attention to recent results that have proven lower bounds for solving LCS under the Strong Exponential Time Hypothesis. We will first showcase two results, one by Abboud et al. (FOCS’15) and the other by Bringmann et al. (FOCS’15), where sub-quadratic lower bounds have been proven for the general LCS problem. Later, we turn to Bringmann et al.’s (SODA’18) which examined the optimal running time of multivariate LCS which proved lower bounds that coincide with the running times of the fastest known algorithms.
  • 27/10/2020
    14:45 - 15:15
    BSc
    Stable and Efficient Algorithms for Logarithmically Counting Homomorphisms from Selected Graph Classes
    Anton Florey
    Graph homomorphisms are mappings between two graphs H and G that preserve all edge relationsof the left-hand side graph. The graph homomorphism counting problem #Hom(H, G) asks for
    the number of homomorphisms from one graph H ∈ H to another graph G ∈ G. It is polynomial
    time solvable for certain left-hand graph classes H. As part of his recent master thesis, Maximilian
    Merz already provided implementations of various homomorphism counting algorithms. Unfor-
    tunately, they quickly get impracticable for large problem instances, as homomorphism numbers
    grow extremely fast. This work revisits most of these algorithms with the novel idea of counting
    the logarithm of these numbers. One main contribution is the implementation and evaluation of
    these adapted algorithms. For most of the newly implemented functions, a significant improve-
    ment in efficiency over the exact counting implementations can be measured. Further experiments
    also show that errors of this logarithmic approximation stay sufficiently close to the unavoidable

    accuracy loss induced by finite machine precision.
    https://rwth.zoom.us/j/95988725888?pwd=NUlaM05VdU5Kdkp0NXVTZHplWlFSQT09
  • 23/10/2020
    10:30 - 11:00
    MSc
    Detecting Substructures with Graph Neural Networks
    Michael Scholkemper
    [master]We present theoretical results relating the computational expressiveness of graph neural networks (GNNs) to the color refinement algorithm for graph isomorphism (Morris et al., X et al.). These yield that given the wrong initialisation, graph neural networks are not able to detect triangles on regular graphs. We introduce a refinement algorithm for graph isomorphism that outperforms color refinement on regular graphs and propose a GNN-architecture motivated by this new algorithm. We experimentally investigate the effect of different initialisations on  the detection of triangles and extend this to k-cycles and k-cliques.
  • 16/10/2020
    10:00 - 11:00
    ATI
    Learning Concepts Described by Weight Aggregation Logic
    Steffen van Bergerem
    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. This is joint work with Nicole Schweikardt.
  • 13/10/2020
    10:00 - 10:30
    BSc
    A comparison of translation from LTL to Büchi automata
    Anna Maiworm
  • 01/10/2020
    10:00 - 10:45
     
    Bachelor Vortrag Jan-Christoph Kassing
    Vortrag zur Bachelorarbeit: The Recursive Algorithm for Parity Games
  • 01/10/2020
    10:00 - 10:45
     
    Bachelor Vortrag Jan-Christoph Kassing
    Vortrag zur Bachelorarbeit: The Recursive Algorithm for Parity Games
  • 29/09/2020
    10:00 - 10:45
     
    Bachelor-Vortrag Johannes Lehmann
    Vortrag zur Bachelorarbeit "Exact Minimization of omega-Automata"
  • 29/09/2020
    10:00 - 10:45
     
    Bachelor-Vortrag Johannes Lehmann
    Vortrag zur Bachelorarbeit "Exact Minimization of omega-Automata"
  • 25/09/2020
    10:30 - 11:00
    MSc
    Master-Vortrag: Graph Autoencoder for Chemical Compounds
    Stefanie Winkler
    https://rwth.zoom.us/j/91919735145?pwd=OWFJMDZ5aTErUjhLWEN0T1AwY0lIdz09 A challenging task in fuel design is to find suitable molecules that optimise desired properties like cetane and octane numbers. Our goal is to automate the generation of molecular graphs whose corresponding molecules serve as components for fuels with a high auto-ignition quality. We use three different neural network models built on variational autoencoders and a generative adversarial network: JT-VAE by Jin et al., MHG-VAE by Kajino and MolGAN by De Cao and Kipf [1–3]. Using a subset of QM9 for training, we adapt the models to optimise different combinations of the cetane and octane numbers of new molecules through Bayesian optimisation. Our results show that all models are able to produce valid molecules with high cetane and octane numbers. MHG-VAE is able to generate most of the highest scoring molecules in the different experiments. Moreover, it creates a lot more unique molecules than the other two models, many of which are easy to synthesise.
  • 25/09/2020
    11:15 - 12:00
    BSc
    Lower Bounds for the Weisfeiler-Leman Dimension of Graphs of Bounded Tree Width
    Lea Schirp
    The Weisfeiler-Leman (WL) algorithm is an iterative approach used to classify graphs and other relational structures. For every natural number k, there is a k-dimensional version, which colors k-tuples of vertices, running in polynomial time but there are also non-isomorphic graphs, which the k-dim version cannot distinguish. This makes it useful to know the WL-dimension of a graph, which is the least natural number k such that the k-dim WL distinguishes the graph from all other non-isomorphic graphs. It is known that bounds on the WL dimension of a graph can be determined if its tree width is known. In this talk, we present tools for computing the tree width of a graph with a focus on Tamaki's implementation. We finally discuss the results of an experimental evaluation of the tree width of these graphs with the goal of finding out if the currently best known bounds on the WL dimension of graphs parametrized by their tree width are tight.
  • 25/09/2020
    12:30 - 13:00
    BSc
    Counting Homomorphisms via Model Counting and Knowledge Compilation
    Patrick Bögel
    The homomorphism counting problem #HOM asks how many homomorphisms there are from a graph H to a graph G. The model counting problem #SAT asks how many satisfying assignments a propositional formula has. Both problems are #P-complete, but #HOM becomes tractable if the left-hand side graphs are restricted to a class of bounded tree-width. In this talk we consider reductions from #HOM to #SAT with special attention to whether the tractable subproblems of #HOM are mapped to tractable subproblems of #SAT. Specifically we find reductions that map homomorphism counting instances where the left-hand side graph is a tree, to formulas which adhere to structural restrictions under which #SAT becomes tractable. We also utilize the approach of knowledge compilation to create formulas in a representation that allows for the model count to be calculated in linear time of its size. We evaluate the reductions practically with multiple state-of-the-art model counters and find the reduction to be inferior to the direct approach.
  • 18/09/2020
    10:00 - 11:00
    ATI
    Tuple-Independent Representations of Probabilistic Databases
    Christoph Standke
    https://rwth.zoom.us/j/96525649453?pwd=bHRma0lmV2xuRTNhWUdQbi9ockdNZz09
  • 03/08/2020
    12:00 - 12:30
    BSc
    Bachelor Kolloquium Meder
  • 28/07/2020
    16:00 - 17:00
     
    Vortrag Wolfgang Gatterbauer
  • 28/07/2020
    16:00 - 17:00
     
    Vortrag Wolfgang Gatterbauer
  • 26/06/2020
    10:00 - 11:00
    ATI
    ATI Seminar Jan Böker
    Jan Böker
  • 19/06/2020
    10:00 - 10:45
    ATI
    ATI Seminar Vortrag Steffen van Bergerem
    Steffen van Bergerem
  • 05/06/2020
    10:00 - 11:00
    ATI
    omega-Word Automata with Global Constraints
    Patrick Landwehr
  • 29/05/2020
    10:00 - 10:30
    ATI
    Graph isomorphism in quasipolynomial time parameterized by treewidth
    Daniel Wiebking
  • 22/05/2020
    10:00 - 11:00
    ATI
    The Iteration Number of Colour Refinement
    Sandra Kiefer
    The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph. A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n ≥ 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation. This is joint work with Brendan McKay.
  • 27/02/2020
    10:00 - 10:30
    MSc
    Unsupervised machine learning for constraint satisfaction problems
    Jan Tönshoff
  • 27/02/2020
    10:00 - 10:30
    MSc
    Unsupervised machine learning for constraint satisfaction problems
    Jan Tönshoff
  • 17/01/2020
    10:00 - 11:00
    ATI
    Generative Datalog with Continuous Distributions
    Peter Lindner
    Probabilistic Databases (PDBs) are a formal model of uncertainty in relational databases, as might occur in a variety of practical application scenarios such as noisy or unreliable input data, data integration or data cleaning. Quite recently, Bárány et al. (TODS 2017) proposed a language called "Probabilistic Programming Datalog (PPDL)" which uses classic Datalog rules that are extended by random sampling. In a nutshell, PPDL is a declarative probabilistic programming language with very close ties to database applications and can be seen as a tool to specify PDBs. In this talk, we focus on the generative part of the language, "Generative Datalog". While the original language of Bárány et al. only supported discrete probability distributions, we allow using probability density functions and inputs that are already PDBs themselves. We present the formal semantics of the language and discuss various properties and consequences, most notably, the support of PDB inputs and robustness with respect to the order of rule applications. This is joint work with M. Grohe, B. L. Kaminski, J.-P. Katoen.
  • 15/01/2020
    10:00 - 10:30
    MSc
    Algorithms for counting homomorphisms from small treewidth graphs
    Maximilian Merz
  • 10/01/2020
    10:00 - 11:00
    ATI
    Isomorphism Testing: From Strings to Hypergraphs
    Daniel Neuen

    We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices $V$ and a permutation group $Gamma$ over domain $V$, and asking whether there is a permutation $gamma in Gamma$ that proves the two hypergraphs to be isomorphic. We show that for input groups, all of whose composition factors are isomorphic to a subgroup of the symmetric group on $d$ points, this problem can be solved in time $(n+m)^{O((log d)^{c})}$ for some absolute constant $c$ where $n$ denotes the number of vertices and $m$ the number of hyperedges. The previous best algorithm for this problem due to Schweitzer and Wiebking (STOC 2019) runs in time $n^{O(d)}m^{O(1)}$.
     
    As an application of this result, we obtain, for example, an algorithm testing isomorphism of graphs excluding $K_{3,h}$ ($h geq 3$) as a minor in time $n^{O((log h)^{c})}$. In particular, this gives an isomorphism test for graphs of Euler genus at most $g$ running in time $n^{O((log g)^{c})}$.
  • 13/12/2019
    10:00 - 11:00
    ATI
    RUN-CSP: Recurrent unsupervised network for CSPs
    Martin Ritzert

    Constraint satisfaction problems form an important and wide class of combinatorial search and optimization problems with many applications in AI and other areas. We introduce a recurrent neural network architecture RUN-CSP (Recurrent Unsupervised Neural Network for Constraint Satisfaction Problems) to train message passing networks solving binary constraint satisfaction problems (CSPs) or their optimization versions (Max-CSP). The architecture is universal in the sense that it works for all binary CSPs: depending on the constraint language, we can automatically design a loss function, which is then used to train generic neural nets. In this paper, we experimentally evaluate our approach for the 3-colorability problem (3-Col) and its optimization version (Max-3-Col) and for the maximum 2-satisfiability problem (Max-2-Sat). We also extend the framework to work for related optimization problems such as the maximum independent set problem (Max-IS). Training is unsupervised, we train the network on arbitrary (unlabeled) instances of the problems. Moreover, we experimentally show that it suffices to train on relatively small instances; the resulting message passing network will perform well on much larger instances (at least 10-times larger).
  • 29/11/2019
    10:00 - 11:00
    ATI
    Fractional Sets, Fractional Decompositions, and Fractional Tangles
    Eva Fluck
  • 29/11/2019
    10:00 - 11:00
    ATI
    Fractional Sets, Fractional Decompositions, and Fractional Tangles
    Eva Fluck
  • 22/11/2019
    10:00 - 11:00
    ATI
    Ambiguity in Probabilistic Büchi Automata
    Anton Pirogov
    Probabilistic Büchi automata are a natural generalization of PFA to infinite words, but have been studied in-depth only rather recently and many interesting questions are still open. PBA are known to accept, in general, a class of languages that goes beyond the regular languages. In this talk I will present new classes of restricted PBA which are still regular, strongly relying on notions concerning ambiguity in classical ω-automata.
  • 20/11/2019
    13:15 - 13:45
    MSc
    Algorithms for Maximum Matching Width
    Yassin Bahloul
  • 20/11/2019
    15:00 - 15:30
    BSc
    The Fine-Grained Complexity of First-Order Properties
    Louis Härtel
    Based on the results by Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova and Ryan Williams, we break down the fine-grained complexity of first-order properties by analyzing their reduction from any model-checking problem on a (k + 1)-quantifier first-order formula to the sparse version of k-Orthogonal Vectors, and summarize their algorithmic improvements and consequences for common fine-grained conjectures.
  • 15/11/2019
    10:00 - 11:00
    ATI
    How Stable are Node Embeddings and does it Matter?
    Hinrikus Wolf
  • 25/10/2019
    10:00 - 11:00
    ATI
    Tree automata with global constraints for infinite trees
    Patrick Landwehr
    We study an extension of tree automata on infinite trees with global equality and disequality constraints. These constraints can enforce that all subtrees for which in the accepting run a state q is reached (at the root of that subtree) are identical, or that these trees differ from the subtrees at which a state q' is reached. We consider the closure properties of this model, its decision problems and its connection to logic.
  • 24/10/2019
    10:00 - 10:30
    BSc
    Regular Sensing for Nested Word Automta
    Alina Ibach
  • 17/10/2019
    10:00 - 10:30
    BSc
    Comparing learning algorithms for regular expressions from positive examples
    Konrad Ostrowski
  • 17/10/2019
    10:45 - 11:15
    BSc
    A comparison of algorithms for automata learning on sparse data
    Caspar Zecha
  • 16/09/2019
    11:00 - 11:30
     
    The Complexity of First-Order Model Checking on Graphs of Bounded Tree Depth
    Jonathan du Mesnil de Rochemont
    In this talk, we discuss algorithmic meta-theorems for graphs of bounded tree-depth. Tree-Depth is a graph invariant also known as vertex ranking number and minimum elimination tree height, which captures, intuitively speaking, how much a graph resembles a star graph. We present a result due to Chen and Flum stating that the model-checking problem for FO parameterized by the length of the formula is in para-AC0 if the inputs are limited to any class of graphs of bounded tree-depth. para-AC0 can be viewed as the complexity class of parameterized problems that are decidable by polynomial-size constant-depth circuit families with arbitrary fan-in after a precomputation on the parameter. An important ingredient in the proof is the following characterization: We show that the model-checking problem for FO on any class of structures is in para-AC0 if and only if FO has an effective generalized quantifier elimination on that class. We then show that FO has such an effective generalized quantifier elimination on classes of graphs of bounded tree-depth to conclude the proof.
  • 16/09/2019
    11:45 - 12:15
     
    Data structures for approximate membership queries
    Razvan Manea
  • 09/09/2019
    14:00 - 15:00
     
    Representations of Correlated Probabilistic Databases
    Nils Freyer
    As an extension of Codd's relational data model, probabilistic databases were discussed in the literature when the relational data model became popular in the 1980's. The studies of probabilistic databases were motivated by errors in the data collection process and designed as a generalisation of the relational data model that is able to integrate probabilistic data into relational data. Today, the amount of data and information is growing strictly, which causes a lot of research in information extraction and efforts in representing the extracted information. One can imagine that representing correlated probabilistic data is not trivial. We will present the most common representation systems for probabilistic data and examine their properties. Afterwards, we will construct translations between the representation systems to draw conclusions on their relationships.
  • 14/06/2019
    10:00 - 10:30
    ATI
    A unifying method for the design of algorithms canonizing combinatorial objects
    Daniel Wiebking

    We devise a unified framework for the design of canonization algorithms. 
    Using hereditarily finite sets, we define a general notion of combinatorial objects 
    that includes graphs, hypergraphs, relational structures, codes, permutation 
    groups, tree decompositions, and so on. 
    Our approach allows for a systematic transfer of the techniques that have been 
    developed for isomorphism testing to canonization. We use it to design a canonization algorithm for general combinatorial objects. This result gives new fastest canonization algorithms with an asymptotic running time matching the best known isomorphism algorithm for the following types of objects: hypergraphs, hypergraphs of bounded color class size, permutation groups (up to permutational isomorphism) 
    and codes that are explicitly given (up to code equivalence). 
  • 31/05/2019
    10:00 - 11:00
    ATI
    The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs
    Daniel Neuen

    The Weisfeiler-Leman procedure is a widely-used approach for graph
    isomorphism testing, which iteratively computes an isomorphism-invariant
    coloring of vertex tuples. Meanwhile, a fundamental tool in structural
    graph theory, which is often exploited in approaches to tackle the graph
    isomorphism problem, is the decomposition into bi- and triconnected
    components.
    We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly
    computes the decomposition of a graph into its triconnected components.
    This means that the decomposition into triconnected components is "for
    free" with respect to the dimension of the algorithm needed to
    distinguish two graphs (assuming dimension at least 2).
    This result implies that the k-dimensional algorithm distinguishes
    k-separators, i.e., k-tuples of vertices that separate the graph, from
    other vertex k-tuples. As a byproduct, we also obtain insights about the
    connectivity of constituent graphs of association schemes.
    In an application of the results, we show the new upper bound of k on
    the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a
    construction by Cai, Fürer, and Immerman, we also provide a new lower
    bound that is asymptotically tight up to a factor of 2.
    This is joint work with Sandra Kiefer.
  • 31/05/2019
    10:00 - 11:00
    ATI
    The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs
    Daniel Neuen

    The Weisfeiler-Leman procedure is a widely-used approach for graph
    isomorphism testing, which iteratively computes an isomorphism-invariant
    coloring of vertex tuples. Meanwhile, a fundamental tool in structural
    graph theory, which is often exploited in approaches to tackle the graph
    isomorphism problem, is the decomposition into bi- and triconnected
    components.
    We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly
    computes the decomposition of a graph into its triconnected components.
    This means that the decomposition into triconnected components is "for
    free" with respect to the dimension of the algorithm needed to
    distinguish two graphs (assuming dimension at least 2).
    This result implies that the k-dimensional algorithm distinguishes
    k-separators, i.e., k-tuples of vertices that separate the graph, from
    other vertex k-tuples. As a byproduct, we also obtain insights about the
    connectivity of constituent graphs of association schemes.
    In an application of the results, we show the new upper bound of k on
    the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a
    construction by Cai, Fürer, and Immerman, we also provide a new lower
    bound that is asymptotically tight up to a factor of 2.
    This is joint work with Sandra Kiefer.
  • 24/05/2019
    10:00 - 11:00
     
    The Complexity of Homomorphism Indistinguishability
    Jan Böker
    We study the complexity of HomInd(F): for a class F of graphs, HomInd(F) denotes the problem of deciding whether two given graphs G and H are homomorphism indistinguishable over F, i.e., whether for every graph F' in F, the number of homomorphisms from F' to G equals the corresponding number from F' to H. We show that there is a polynomial-time decidable class F of graphs of bounded treewidth for which HomInd(F) is undecidable. Our second hardness result concerns the class K of complete graphs: We show that HomInd(K) is coNP-hard. In fact, we show that it is complete for the class C=P and, hence, apparently much harder than coNP. We conclude our studies of HomInd(F) with a tractability result: HomInd(P) can be solved in polynomial time for the class P of directed paths. Finally, we briefly study some variants of HomInd(F). This is joint work with Yijia Chen, Martin Grohe, and Gaurav Rattan.
  • 23/05/2019
    13:00 - 14:00
     
    The Weisfeiler-Leman Dimension of Graphs of Bounded Treedepth
    Luca Oeljeklaus
    In this talk we discuss the dimension of the Weisfeiler-Leman (WL) algorithm, which is a combinatorial algorithm used as a subroutine for graph isomorphism testing, in terms of treedepth, a graph invariant also referred to as minimum elimination tree height or vertex ranking number. More precisely, we prove upper and lower bounds on the WL-dimension of graphs of bounded treedepth by using two results from Cai, Fürer, and Immerman. In our proof to show that every graph of treedepth at most k is identified by the (k-1)-dimensional WL-algorithm, we use the fact that the (k-1)-dimensional WL-algorithm identifying a graph is equivalent to Spoiler having a winning strategy for the C_k-pebble game between a graph of treedepth at most k and any other non-isomorphic graph. From there we develop an inductive winning strategy for Spoiler over the treedepth of the graph. In our proof to show that there exists a family of pairs of graphs of treedepth k for arbitrarily large k such that the (k/25)-dimensional WL-algorithm does not distinguish them, we apply the CFI-graph construction to a family of graphs with large separators and then prove an upper bound on the treedepth of the resulting pairs of graphs.