
Date
Title

08/02/2019
10:00  11:00
ATI
The WLdimension of graphs of bounded rank widthDaniel NeuenWe prove that the combinatorial WeisfeilerLeman algorithm of dimension (3k+4) is a complete isomorphism test for the class of all graphs of rank width at most k. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width k is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time n^{f(k)} for a nonelementary function f. Our result yields an isomorphism test for graphs of rank width k running in time n^{O(k)}. Another consequence of our result is the first polynomial time canonisation algorithm for graphs of bounded rank width. If time permits I will also speak about our second result that fixedpoint logic with counting captures PTIME on the class of graphs of rank width at most k.

24/01/2019
10:00  10:30
BSc
A neural algorithm for similarity search
Christian Blumenthal

24/01/2019
11:00  11:45
MSc
State Space Reduction For Parity Automata
Andreas Tollkötter

18/01/2019
10:00  11:00
ATI

11/01/2019
10:00  11:00
ATI
On the WLdimension of graphs of bounded rank widthDaniel Neuen

14/12/2018
10:00  11:00
ATI
Color Refinement and Graph Neural NetworksMartin RitzertIn this talk we show that the expressive power of the 1dimensional Weisfeiler Leman algorithm (color refinement) is exactly as powerful as the emerging graph neural networks. Joint work with Martin Grohe and Gaurav Rattan

12/12/2018
11:00  11:45
BSc
Spektrale Graphähnlichkeit und Dominanz
Athena Riazsadri
Im Rahmen dieser Arbeit werden die Probleme der spektralen Graphdominanz (GD) und des spektralen Graphähnlichkeit (SGS) vorgestellt. Anstatt etwa die Anzahl der nicht gematchten Kanten zu vergleichen, werden Graphen aufgrund ihrer Eigenwerte und Laplacematrizen verglichen. Damit bietet die hier vorgestellte Lösung im Gegensatz zu anderen Lösungsansätzen zum Graphähnlichkeitsproblem die Möglichkeit, Graphen aufgrund ihrer Funktionalität zu vergleichen, was wiederum Anwendung gerade in der Biologie (Proteinnetzwerke) finden könnte. Die Hauptergebnisse dieser Arbeit sind ein NPSchwereBeweis für GD und ein κ^4Approximationsalgorithmus für SGS auf zwei Graphen beschränkten Grades. Dieser Algorithmus läuft in polynomieller Zeit für ein konstantes κ.

07/12/2018
10:00  11:00
ATI

30/11/2018
10:00  11:00
ATI
Universal graphs for parity and meanpayoff gamesNathanaël FijalkowI will present a new tool for understanding games on graphs called universal graphs. They can be used to construct algorithms for solving games such as parity games and meanpayoff games. The goals of this talk are to: * introduce the notion of universal graphs and their applications to games * give a simple and unified presentation of all three quasipolynomial time algorithms for parity games * construct new algorithms for meanpayoff games * show tight lower bounds for the construction of algorithms in this framework for both parity and meanpayoff games

30/11/2018
10:00  11:00
ATI
Universal graphs for parity and meanpayoff gamesNathanaël FijalkowI will present a new tool for understanding games on graphs called universal graphs. They can be used to construct algorithms for solving games such as parity games and meanpayoff games. The goals of this talk are to: * introduce the notion of universal graphs and their applications to games * give a simple and unified presentation of all three quasipolynomial time algorithms for parity games * construct new algorithms for meanpayoff games * show tight lower bounds for the construction of algorithms in this framework for both parity and meanpayoff games

23/11/2018
10:30  11:30
ATI
Detecting small subgraphs using the exterior algebraHolger DellColorcoding is a popular technique to detect and to approximately count short paths and other subgraphs of bounded treewidth. In this talk, we will discuss Extensorcoding, a natural approach to such subgraph detection problems that turns out to generalize and in some cases improve upon the running time achieved by Colorcoding. Based on joint work with Cornelius Brand and Thore Husfeldt.

16/11/2018
10:00  11:00
ATI
Structural Similarity and Homomorphism CountsJan Böker

10/11/2018
00:00  03:00
Wissenschaftsnacht der Professoren mit DJ Prof. Grohe
DJ Prof. Grohe

09/11/2018
10:30  11:30
ATI
A unifying method for the design of algorithms canonizing combinatorial objectsDaniel WiebkingWe 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).

26/10/2018
14:15  15:15
Explaining and Representing Novel Concepts With Minimal SupervisionZeynep Akata (University of Amsterdam)Clearly explaining a rationale for a classification decision to an enduser can be as important as the decision itself. Existing approaches for deep visual recognition are generally opaque and do not output any justification text; contemporary visionlanguage models can describe image content but fail to take into account classdiscriminative image properties which justify visual predictions. In this talk, I will present my past and current work on ZeroShot Learning, Vision and Language for Generative Modeling and Explainable Artificial Intelligence where we show (1) how to generalize image classification models to cases when no visual training data is available, (2) how to generate images and image features using detailed visual descriptions, and (3) how our models focus on discriminating properties of the visible object, jointly predict a class label, explain why/not the predicted label is chosen for the image.

23/10/2018
16:15  17:00
Polyregular functionsMikolaj BojanczykUnRAVeL guest talk Date: Tuesday, 23. October, at 16:15 Location: Room 9U10 (Building E3) Speaker: Mikolaj Bojanczyk from University of Warsaw, Poland Title: Polyregular functions Abstract: I will describe a class of stringtostring transducers, which goes beyond rational or regular functions, but still shares many of their good properties (e.g. the inverse image of a regular language is regular). Unlike many stringtostring transducer models (including sequential, rational and regular transducers), which have linear size increase, the the new class (called polyregular functions) has polynomial size increase. The main selling point of the polyregular functions is that they admit numerous equivalent descriptions: ￼ automata with pebbles; (b) a fragment of lambda calculus; (c) a fragment of forprograms; and (d) compositions of certain atomic functions, such as reverse or a form of string squaring. Also, most likely (this is still ongoing work): (e) mso stringtostring interpretations.

10/10/2018
10:00  10:45
MSc
Spectral graph similarity
Timo Gervens

05/10/2018
11:00  11:45
BSc
Weighted Symmetric Model Counting for Two Variable Logic
Tobias Schleifstein
The first order model counting problem (FOMC) asks for the number of models of a FOsentence with the domain of given size n. The weighted first order model counting problem (WFOMC) additionally assigns a weight to every tupel of the models' relations. In the symmetric variant, which is mainly studied in this thesis, we have to assign the same weights to all tupels of the same relation. We review the results for tractability of the symmetric WFOMC for twovariable logic and threevariable logic. In particular we discuss the data complexity, i.e., the complexity of computing the problems with only the size n of the domain as an input and the formula and weights fixed. It has been shown that the WFOMC is computable in polynomial time for FO² and an extension, but #P_1complete for FO³.

28/09/2018
10:15  11:00
BSc
Algorithms for the Maximum Common Subtree Isomorphism Problem
Florian Frantzen
The maximum common subtree isomorphism problem asks for the largest possible isomorphism among all common subtree isomorphisms between two unrooted input trees. This can either be formulated in terms of the size of the common subtree, or more generally with respect to a weight function. Its more general formulation of a maximum common subgraph isomorphism problem, which permits arbitrary input graphs instead of only trees, is known to be NPcomplete. The restricted setting allows for efficient polynomial time algorithms. We present a particular recent algorithm due to Droschinsky, Kriege and Mutzel for this problem. For unrooted trees of order n and degree Delta, this algorithm achieves a worstcase running time of O(n^2 Delta). We complement this theoretical description of the algorithm with various tests of possible changes to the algorithm and compare their impacts on our own implementation of the algorithm. We also compare our implementation to other programs that can be used to compute a solution for the maximum common subtree isomorphism problem.

28/09/2018
11:00  11:45
BSc
The quantifier depth for distinguishing structures in firstorder logic
Bianka Bakullari
The WeisfeilerLeman algorithm is a combinatorial procedure used to distinguish nonisomorphic graphs. I give an introduction to the mechanisms of the algorithm by putting the focus on the 2dimensional variant of it, for which I explain the nontrivial upper bound O(n^2/log n) on the number of iterations. Then graphs are defined as relational structures to show the connection between the dimension of the algorithm and the number of variables used in logics with counting. In the end, we show an approach to represent arbitrary finite relational structures as colored graphs in a way that we can apply the WeisfeilerLeman algorithm to distinguish them. We use the connection between the algorithm and logics with counting to make claims about the quantifier depth of the formulas that distinguish nonisomorphic finite relational structures.

28/09/2018
11:45  12:30
BSc
Parameterized Approximability of Dominating Set
Daniel Schleiz
Due to the NPhardness of the Dominating Set decision problem, there is no efficient algorithm solving it unless P=NP. Seperate approximation and parameterization approaches to make this problem more tractable have shown to yield dissatisfactory results. Thus the parameterized approximability of Dominating Set has been an open question for a while, where for a Graph G with n vertices and an integer k given as input, an algorithm computing a dominating set of size at most H(k)*k with a runtime of F(k)*poly(n) for some computable functions H,F is of interest. Only recently Karthik, Laekhanukit and Manurangsi (2018) were able to rule out such an algorithm under the assumption of W[1]!=FPT. In this bachelor thesis talk, the proof for this result will be presented along with the needed background on parameterized complexity. Furthermore, a stronger runtime lower bound of F(k)*n^{o(k)} up to the same approximation ratio assuming the Exponential Time Hypothesis will be shown, resulting from the same work of Karthik et al.

28/09/2018
12:30  13:15
BSc
Optimizationbased Hierarchical Clustering
Michael Scholkemper
Hierarchical clustering is a data analysis method that recursively partitions a given data set into increasingly smaller clusters. The structure that is obtained from this procedure is then represented by a tree  or a dendogram. This thesis presents the work by [Dasgupta, 2016] and the subsequent work by [CohenAddad et al., 2018] that spawned the theoretical study of the problem. Therein an optimization framework for Hierarchical Clustering is established by defining an objective function, thus framing it as a combinatorial optimization problem. This thesis presents the needed theory for the proposed optimization framework and provides approximation algorithms for both similarity and dissimilarity inputs as. Additionally, a family of inputs constrained in such a way as to admit efficient Hierarchical Clustering is presented.

27/09/2018
14:00  14:45
MSc
Optimization for the complementation of Büchi automata
Lasse Nitz

26/09/2018
11:00  12:00
MSc
Realvalued Connectivity Systems
Eva Fluck
Based on a survey on connectivity, decompositions and tangles by Grohe [5], we adapt the so far integervalued theory onto realvalued set functions. We start by adapting the definition of finite connectivity systems including the def inition of connectivity functions and give examples of different functions and uni verses. A connectivity function is a set function κ : 2 U → R on a finite universe U , which is symmetric, normalized and submodular, that is, for all X, Y ∈ U , κ(X)+κ(Y ) ≥ κ(X ∩Y )+κ(X ∪Y ). We find that the codomain of κ is discrete. This allows us, similar to integervalues, to do inductive arguments using a value called granularity, which is the smallest nonzero difference between two function values. Then we look at the definitions of tangles, which we view as highly connected regions of the universe, and branch decompositions, which are tree like decompositions of the universe related to the wellknown tree decompositions, into this new setting. We are able to prove that both definitions behave similar to integervalued systems. Next we derive the duality theory between branch decompositions and tangles, first developed by Robertson and Seymour [11]. This duality states, that we can construct a branch decomposition of a certain width k if there is no tangle of order k and that a branch decomposition of width k is a witness for the nonexistence of a tangle of order k. So far this duality was mostly based on the submodularity of the set function. We find additional functions, where branch decomposition is also dual to tangles. We find a property of the set function that we call submodular bounded, that is, for all sets X, Y ∈ U and all k ∈ R, if κ(X), κ(Y ) ≤ k then κ(X ∩ Y ), κ(X ∪ Y ) ≤ k. Submodular bounded leads to an equivalent duality theory as submodularity. Lastly, we show that there is a way to canonically decompose realvalued connectivity systems into their maximal tangles.

26/09/2018
12:00  12:45
BSc
DistanceAware Colour Refinement
Björn Plewinski
Color refinement is a graph algorithm used in isomorphism testing which also shows great promise in designing graph kernels for machine learning applications. This bachelor thesis explores an extension of this algorithm which incorporates distances between vertices. We will examine the algorithms theoretical properties and compare it and its performance with the standard color refinement algorithm and the twodimensional WeisfeilerLeman algorithm.

25/09/2018
10:00  10:30
BSc
Nondeterministic automata with deterministic acceptance strategies
Domenic Quirl

25/09/2018
10:45  11:15
BSc
Canonical automata for regular languages
Magnus Groß

14/09/2018
11:00  12:00
MSc
Structural Similarity and Homomorphism Counts
Jan Böker
Recent results show how the structural similarity of graphs can be characterized by counting homomorphisms to them. We show how homomorphism counts characterize the structural similarity of related relational structures, with a focus on directed graphs, (vertex)colored graphs, and multihypergraphs. We prove that two directed graphs G and H are isomorphic if and only if, for every directed acyclic graph D, the number of homomorphisms Hom(D,G) from D to G is equal to the corresponding number Hom(D,H) from D to H. Our approach also yields a similar statement for undirected graphs, where it suffices to count homomorphisms from threecolorable graphs. We show how, after encoding a colored graph as an uncolored one, counts of homomorphisms to these graphs can be obtained from each other. This allows us to generalize a recent result on uncolored graphs, obtaining that two colored graphs G and H are not distinguished by the colorrefinement algorithm if and only if, for every colored tree T, we have Hom(T, G) = Hom(T, H). Additionally, this reduction also works for a generalization of said result to bounded treewidth. We introduce a generalization of the colorrefinement algorithm for multihypergraphs and prove that it does not distinguish two multihypergraphs G and H if and only if, for every Bergeacyclic multihypergraph B, we have Hom(B, G) = Hom(B, H). To this end, we show how homomorphisms of multihypergraphs and their colored incidence graphs are related to each other.

14/09/2018
12:00  12:45
BSc
EndtoEnd Graph Learning
Emre Yamen
In der Präsentation wird EndtoEnd Graph Learning definiert, was eine Möglichkeit der Graphen Klassifizierung ist. Zudem wird erklärt wie man Graphen kanonisieren kann. Danach wird eine aktuelle Methode zur Graphen Klassifizierung präsentiert, die Graph Kernel. Da wird mehr auf die WeisfeilerLeman subtree kernel eingegangen. Danach werden Experimente zur Graphen Klassifizerung mit EndtoEnd Graph Learning und Graph Kanonisierung in einem neuronalen Netzwerk vorgestellt. Das Ergebnis wird mit dem Ergebnis vom WeisfeilerLeman subtree kernel in Neuronalen Netzwerken verglichen.

05/09/2018
09:30  10:30
BSc
Similarity of Small Graphs
Stefanie Winkler
The graph similarity problem deals with the computation of the distance between two given graphs and is therefore closely related to graph isomorphism. There are different approaches to define this distance. This thesis mainly deals with the Frobenius distance. Besides proving the NPcompleteness of the problem, we examine its behavior on small graphs. To achieve this, we implement it as a mathematical optimization model in Java using the tool CPLEX. We use the implementation to compute the distance of all graphs up to size five and for some selected bigger graphs. To summarize the results, we present them in multiple plots showing the occurring distances and the runtime. Since the problem is NPhard, it is not efficient to solve unless P = NP. Therefore, we approximate the exact calculation by convex relaxation. We examine the resulting measure as done for the unrelaxed case. Furthermore, we directly compare the computed distances on pairs of graphs to analyze the ratio between the two. Other approaches to graph similarity include the minimum graph edit distance and the maximum common subgraph. We present their respective definition and prove their NPhardness.

29/08/2018
10:00  11:00
BSc
Graph Autoencoder
Florian Behrens

29/08/2018
10:00  11:00
BSc
Graph Autoencoder
Florian Behrens

24/08/2018
10:00  10:30
Climbing up the elementary complexity classes with theories of automatic structuresPeter LindnerAutomatic structures are of particular interest, since their first order theories are decidable. While these theories might be of nonelementary complexity in general, the practically relevant examples are located below 3fold exponential time (e. g. Presburger arithmetic and automatic structures of bounded degree). It was an open question [DurandGasselin], whether there are automatic structures with theories of higher elementary complexity. In my masters thesis, I introduced a family of automatic structures, roughly covering all of the complexity classes kEXPTIME for k > 2. Presented in this talk is now the joint work with Faried Abu Zaid and Dietrich Kuske (TU Ilmenau), who were able to close the remaining gaps and thus give an exact characterization of the complexity of the mentioned structures. The presented proofs contain various involved encoding techniques.

16/08/2018
15:00  16:00
MSc
Titel: Colour Refinement and Fractional Isomorphism on Weighted Structures
Hinrikus Wolf
Colour refinement is a basic algorithm for graph isomorphism testing. It splits the vertex set of a graph until all vertices in a class have the same number of neighbours in every other class. The resulting partition is called (coarsest) equitable partitionof a graph. Tinhofer, Ramana et al. and Godsil found a tight connection between equitable partitions of a graph and fractional automorphisms, which are the LP relaxation of the natural ILP formulation of graph isomorphism. Grohe et al. generalised the colour refinement algorithm for weighted bipartite graphs in a matrix representation and they introduced a theory of equitable partitions, fractional automorphisms and fractional isomorphisms for weighted bipartite graphs in a matrix representation. We develop a series of generalisations and extensions of their theory. Firstly, we modify the colour refinement algorithm to work on weighted directed graphs in matrix representation. Furthermore, we extend the theory of equitable partitions, fractional automorphisms and fractional isomorphisms to weighted directed graphs. Secondly, we adapt the results for graphs with an additional weighted unary or binary relation. Then we adapt Atserias’ and Maneva’s definition of kequitable partitions to state the kdimensional colour refinement algorithm for weighted directed graphs. Thirdly, we generalise kequitable partitions and kdimensional colour refinement to weighted τstructures. In the end, we state a general theorem on the consistent theory of equitable partitions, fractional automorphisms and fractional isomorphisms.

15/06/2018
10:15  11:15
A unified approach to Büchi determinisationAnton PirogovNondeterministic Büchi automata are one of the simplest types of finite automata on infinite words and are successfully applied in model checking. In some settings nondeterminism is not feasible in practice, but determinisation of Büchi automata is a difficult problem with a long history and a few different solutions. In our work we noticed similarities in two different determinisation constructions and studied those to obtain a generalized algorithm from which both original constructions can be recovered as special cases. In this talk I will sketch both constructions, illuminate their relationship and give an idea of the unified procedure. (Joint work with Christof Löding)

25/05/2018
10:15  11:15
Obstacles for better algorithms for the Graph Isomorphism ProblemDaniel NeuenI will mainly talk about a new algorithm solving the isomorphism problem for graphs of maximum degree d in time n^{polylog(d)} where n denotes the number of vertices of the input graphs. The previous best algorithm for this problem due to Luks runs in time n^{O(d)}. This result also improves on the recent quasipolynomial time algorithm for the general graph isomorphism problem due to Babai and in particular results in a faster algorithm for graphs of logarithmic degree, one of the bottleneck cases in Babai's algorithm. If time permits I will also discuss further bottleneck cases for Babai's algorithm and present some open questions regarding these cases.

18/05/2018
10:15  11:15
Tree Automata with Global ConstraintsPatrick LandwehrWe study an extension of standard tree automata that allows for testing (dis)equality of subtrees, which may be arbitrary far away from each other. We introduce a general model, look at known results on finite trees and discuss where these results can be extended to infinite trees. In particular, we briefly tackle closure properties and then analyze the decidability of the universality and emptinessproblem.

04/05/2018
10:15  11:15
Lovasz meets Weisfeiler and LemanMartin GroheI will speak about an unexpected correspondence between a beautiful theory, due to Lovasz, about homomorphisms and graph limits and a popular heuristic for the graph isomorphism problem known as the WeisfeilerLeman algorithm. I will also relate this to graph kernels in machine learning. Indeed, the context of this work is to desgin and understand similarity measures between graphs and discrete structures. (Joint work with Holger Dell and Gaurav Rattan.)

04/05/2018
10:15  11:15
Lovasz meets Weisfeiler and LemanMartin GroheI will speak about an unexpected correspondence between a beautiful theory, due to Lovasz, about homomorphisms and graph limits and a popular heuristic for the graph isomorphism problem known as the WeisfeilerLeman algorithm. I will also relate this to graph kernels in machine learning. Indeed, the context of this work is to desgin and understand similarity measures between graphs and discrete structures. (Joint work with Holger Dell and Gaurav Rattan.)

19/04/2018
10:30  11:30
Algorithmic Approaches to Testing Isomorphism of Graphs of Bounded Treewidth
Lisa Mannel
The graph isomorphism problem is the problem of deciding for two given graphs whether there exists a bijection between their vertices that preserves the edge relation. It is one of the classical problems in NP for which we do not know whether it is in P. However, many results have been found on the complexity of certain graph classes, for example for graphs of bounded treewidth. We develop a new algorithm which decides isomorphism for two graphs given together with their isomorphisminvariant nested tree decompositions. It employs a subroutine GIT (short for Graph Isomorphism Test), which can be any algorithm deciding isomorphism for colored graphs. The running time of our algorithm heavily depends on the running time of GIT and the properties of the given isomorphisminvariant nested tree decompositions. Our second contribution is the collection and combination of existing algorithms to compute an algorithm that outputs an isomorphisminvariant nested tree decomposition of a given graph. The combination of this algorithm with our approach to testing isomorphism introduced above, which employs Babai's quasipolynomialtime algorithm as the subroutine GIT, allows us to decide isomorphism for graphs of treewidth bounded by k in time 2^{O((k^2 log k)^c)}*p(n), for some polynomial p and a constant c.

26/01/2018
10:15  11:15
ATI
SMTbased Flat Model Checking for LTL with Counting
Anton Pirogov

23/01/2018
10:00  11:00
Provenance for Logics with Team Semantics
Lukas Huwald
Master Kolloquium Abstract: Provenance analysis is a concept that originated in the field of database theory which unifies several nonstandard semantics for database queries, such as bag semantics or probabilistic semantics. These different semantics can be captured in an algebraic framework using various semirings, and are special instances of computations in a general provenance semiring. While the previous work on provenance semantics focused mainly on positive query languages for databases, Grädel and Tannen recently introduced provenance semantics for logical languages that also include negation, e.g. full first order logic. In this master thesis talk we introduce semiring semantics and provenance analysis for logics with team semantics, such as Väänänen's dependence logic. The main features of these logics are atomic formulae that allow reasoning about dependence and independence relations between variables. We define semiring semantics for these logics and analyze their properties, e.g. with respect to locality, closure properties and game theoretic semantics. We show that in the class of idempotent semirings, many of the known results for booolean semantics are preserved when considering semiring semantics. This is not the case for general semirings: many logics that are equivalent for boolean semantics can be separated for semiring semantics. In particular we show that independence logic and existential second order logic are incomparable for semiring semantics, but certain extensions of these logics remain equivalent.

19/01/2018
10:15  11:15
ATI
Theories of Automatic Structures within the Exponential Time Hierarchy
Peter Lindner

22/12/2017
10:15  11:15
ATI
Capacity of Neural Net works for Lifelong Learn ing of Composable Tasks
work by Leslie G. Valiant
Speaker: Dominik Meier

18/12/2017
10:00  10:30
Learning Algorithms for Sequential Transducers
Tom Biskup
Bachelor talk

15/12/2017
10:15  11:15
ATI
Learning Symbolic Automata
work by Samuel Drews and Loris D'Antoni
Speaker: Leo Krömker

08/12/2017
10:15  11:15
ATI
Set Similarity Search Beyond MinHash
work by Tobias Christiani, Rasmus Pagh
Speaker: Lukas Tiago Bolte

17/11/2017
10:15  11:15
ATI
Isomorphism of bounded degree graphs
Daniel Neuen
something with isomorphism and bounded degree

06/11/2017
14:00  15:00
The parameterized complexity of counting perfect matchings and other subgraphs
Radu Curticapean
Counting problems became a part of TCS at least since Valiant introduced the class #P and proved that counting perfect matchings is #Pcomplete. Polynomialtime algorithms for exact counting problems tend to be rare and curious, but even for the #Phard problems, some progress can be made through the usual relaxations studied in TCS. Among those, this talk will cover the parameterized complexity of counting problems, as introduced by Flum and Grohe, and independently by McCartin. One part of the talk will focus on the complexity of counting perfect matchings, a problem studied in algebra, combinatorics, and statistical physics. We discuss how this problem reacts to structural parameters from graph minor theory like the genus, the apex number and the Hadwiger number. This contains some algorithmic results, but also conditional lower bounds under the exponentialtime hypothesis ETH and its stronger sibling SETH. We also consider the problems of counting general subgraphs H in a host graph G, this time parameterized by the size of H. This gives rise to the problems #Sub(C) for fixed graph classes C: Given inputs H and G, where H is from C, count the Hcopies in G. We show a dichotomy theorem for these problems, together with recently found connections to a framework developed by Lovász. The talk is based on several works, some of which involved Holger Dell, Dániel Marx, and Mingji Xia.

03/11/2017
10:15  11:15