arXiv subjects

Martin Grohe

Publications and source records attributed to Martin Grohe.

At least 73 records · Page 4Linked to original sources

arXiv2021

The Surprising Power of Graph Neural Networks with Random Node Initialization

Graph neural networks (GNNs) are effective models for representation learning on relational data. However, standard GNNs are limited in their expressive power, as they cannot distinguish graphs beyond the capability of the Weisfeiler-Leman graph isomorphism heuristic. In order to break this expressiveness barrier, GNNs have been enhanced with random node initialization (RNI), where the idea is to train and run the models with randomized initial node features. In this work, we analyze the expressive power of GNNs with RNI, and prove that these models are universal, a first such result for GNNs not relying on computationally demanding higher-order properties. This universality result holds even with partially randomized initial node features, and preserves the invariance properties of GNNs in expectation. We then empirically analyze the effect of RNI on GNNs, based on carefully constructed datasets. Our empirical findings support the superior performance of GNNs with RNI over standard GNNs.

cs.LG↗

arXiv2021

Probabilistic Data with Continuous Distributions

Statistical models of real world data typically involve continuous probability distributions such as normal, Laplace, or exponential distributions. Such distributions are supported by many probabilistic modelling formalisms, including probabilistic database systems. Yet, the traditional theoretical framework of probabilistic databases focusses entirely on finite probabilistic databases. Only recently, we set out to develop the mathematical theory of infinite probabilistic databases. The present paper is an exposition of two recent papers which are cornerstones of this theory. In (Grohe, Lindner; ICDT 2020) we propose a very general framework for probabilistic databases, possibly involving continuous probability distributions, and show that queries have a well-defined semantics in this framework. In (Grohe, Kaminski, Katoen, Lindner; PODS 2020) we extend the declarative probabilistic programming language Generative Datalog, proposed by (Bárány et al.~2017) for discrete probability distributions, to continuous probability distributions and show that such programs yield generative models of continuous probabilistic databases.

cs.DB↗

arXiv2021

Definable decompositions for graphs of bounded linear cliquewidth

We prove that for every positive integer k, there exists an MSO_1-transduction that given a graph of linear cliquewidth at most k outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of k. A direct corollary of this result is the equivalence of the notions of CMSO_1-definability and recognizability on graphs of bounded linear cliquewidth.

cs.LO↗

arXiv2021

Recent Advances on the Graph Isomorphism Problem

We give an overview of recent advances on the graph isomorphism problem. Our main focus will be on Babai's quasi-polynomial time isomorphism test and subsequent developments that led to the design of isomorphism algorithms with a quasi-polynomial parameterized running time of the from $n^{\text{polylog}(k)}$, where $k$ is a graph parameter such as the maximum degree. A second focus will be the combinatorial Weisfeiler-Leman algorithm.

cs.DS↗

arXiv2020

Database Repairing with Soft Functional Dependencies

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 this problem in 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.

cs.DB↗

arXiv2020

The Effects of Randomness on the Stability of Node Embeddings

We systematically evaluate the (in-)stability of state-of-the-art node embedding algorithms due to randomness, i.e., the random variation of their outcomes given identical algorithms and graphs. We apply five node embeddings algorithms---HOPE, LINE, node2vec, SDNE, and GraphSAGE---to synthetic and empirical graphs and assess their stability under randomness with respect to (i) the geometry of embedding spaces as well as (ii) their performance in downstream tasks. We find significant instabilities in the geometry of embedding spaces independent of the centrality of a node. In the evaluation of downstream tasks, we find that the accuracy of node classification seems to be unaffected by random seeding while the actual classification of nodes can vary significantly. This suggests that instability effects need to be taken into account when working with node embeddings. Our work is relevant for researchers and engineers interested in the effectiveness, reliability, and reproducibility of node embedding approaches.

cs.LG↗

arXiv2020

word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data

Vector representations of graphs and relational structures, whether hand-crafted feature vectors or learned representations, enable us to apply standard data analysis and machine learning techniques to the structures. A wide range of methods for generating such embeddings have been studied in the machine learning and knowledge representation literature. However, vector embeddings have received relatively little attention from a theoretical point of view. Starting with a survey of embedding techniques that have been used in practice, in this paper we propose two theoretical approaches that we see as central for understanding the foundations of vector embeddings. We draw connections between the various approaches and suggest directions for future research.

cs.LG↗

arXiv2020

Deep Weisfeiler Leman

We introduce the framework of Deep Weisfeiler Leman algorithms (DeepWL), which allows the design of purely combinatorial graph isomorphism tests that are more powerful than the well-known Weisfeiler-Leman algorithm. We prove that, as an abstract computational model, polynomial time DeepWL-algorithms have exactly the same expressiveness as the logic Choiceless Polynomial Time (with counting) introduced by Blass, Gurevich, and Shelah (Ann. Pure Appl. Logic., 1999) It is a well-known open question whether the existence of a polynomial time graph isomorphism test implies the existence of a polynomial time canonisation algorithm. Our main technical result states that for each class of graphs (satisfying some mild closure condition), if there is a polynomial time DeepWL isomorphism test then there is a polynomial canonisation algorithm for this class. This implies that there is also a logic capturing polynomial time on this class.

cs.LO↗

arXiv2020

Counting Bounded Tree Depth Homomorphisms

We prove that graphs G, G' satisfy the same sentences of first-order logic with counting of quantifier rank at most k if and only if they are homomorphism-indistinguishable over the class of all graphs of tree depth at most k. Here G, G' are homomorphism-indistinguishable over a class C of graphs if for each graph F in C, the number of homomorphisms from F to G equals the number of homomorphisms from F to G'.

cs.LO↗

arXiv2020

Graph Neural Networks for Maximum Constraint Satisfaction

Many combinatorial optimization problems can be phrased in the language of constraint satisfaction problems. We introduce a graph neural network architecture for solving such optimization problems. The architecture is generic; it works for all binary constraint satisfaction problems. Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger). We experimentally evaluate our approach for a variety of problems, including Maximum Cut and Maximum Independent Set. Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific problems.

cs.AI↗

arXiv2020

Infinite Probabilistic Databases

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.

cs.DB↗

arXiv2019

A Faster Isomorphism Test for Graphs of Small Degree

In a recent breakthrough, Babai (STOC 2016) gave a quasipolynomial time graph isomorphism test. In this work, we give an improved isomorphism test for graphs of small degree: our algorithms runs in time $n^{O((\log d)^{c})}$, where $n$ is the number of vertices of the input graphs, $d$ is the maximum degree of the input graphs, and $c$ is an absolute constant. The best previous isomorphism test for graphs of maximum degree $d$ due to Babai, Kantor and Luks (FOCS 1983) runs in time $n^{O(d/ \log d)}$.

cs.DS↗

arXiv2019

A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded Genus

The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be characterised in terms of the number of variables that is required to describe the graph up to isomorphism in first-order logic with counting quantifiers. It is known that the WL dimension is upper-bounded for all graphs that exclude some fixed graph as a minor (Grohe, JACM 2012). However, the bounds that can be derived from this general result are astronomic. Only recently, it was proved that the WL dimension of planar graphs is at most 3 (Kiefer, Ponomarenko, and Schweitzer, LICS 2017). In this paper, we prove that the WL dimension of graphs embeddable in a surface of Euler genus $g$ is at most $4g+3$. For the WL dimension of graphs embeddable in an orientable surface of Euler genus $g$, our approach yields an upper bound of $2g+3$.

cs.DM↗

arXiv2019

Probabilistic Databases with an Infinite Open-World Assumption

Probabilistic databases (PDBs) introduce uncertainty into relational databases by specifying probabilities for several possible instances. Traditionally, they are finite probability spaces over database instances. Such finite PDBs inherently make a closed-world assumption: non-occurring facts are assumed to be impossible, rather than just unlikely. As convincingly argued by Ceylan et al. (KR '16), this results in implausibilities and clashes with intuition. An open-world assumption, where facts not explicitly listed may have a small positive probability can yield more reasonable results. The corresponding open-world model of Ceylan et al., however, assumes that all entities in the PDB come from a fixed finite universe. In this work, we take one further step and propose a model of "truly" open-world PDBs with an infinite universe. This is natural when we consider entities from typical domains such as integers, real numbers, or strings. While the probability space might become infinitely large, all instances of a PDB remain finite. We provide a sound mathematical framework for infinite PDBs generalizing the existing theory of finite PDBs. Our main results are concerned with countable, tuple-independent PDBs; we present a generic construction showing that such PDBs exist in the infinite and provide a characterization of their existence. This construction can be used to give an open-world semantics to finite PDBs. The construction can also be extended to so-called block-independent-disjoint probabilistic databases. Algorithmic questions are not the focus of this paper, but we show how query evaluation algorithms can be lifted from finite PDBs to perform approximate evaluation (with an arbitrarily small additive approximation error) in countably infinite tuple-independent PDBs.

cs.DB↗

arXiv2018

Colouring and Covering Nowhere Dense Graphs

It was shown by Grohe et al. that nowhere dense classes of graphs admit sparse neighbourhood covers of small degree. We show that a monotone graph class admits sparse neighbourhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak colouring numbers. The core results of this paper are various lower and upper bounds on the weak colouring numbers and other, closely related generalised colouring numbers. We prove tight bounds for these numbers on graphs of bounded tree width. We clarify and tighten the relation between the expansion (in the sense of "bounded expansion" as defined by Nesetril and Ossona de Mendez) and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the generalised colouring numbers. Finally, we show that computing weak r-colouring numbers is NP-complete for all r>2.

math.CO↗

arXiv2018

Lovász Meets Weisfeiler and Leman

In this paper, we relate a beautiful theory by Lovász with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k-dimensional generalization known as the Weisfeiler-Leman algorithm. We prove that two graphs G and H are indistinguishable by the color refinement algorithm if and only if, for all trees T, the number Hom(T,G) of homomorphisms from T to G equals the corresponding number Hom(T,H) for H. There is a natural system of linear equations whose nonnegative integer solutions correspond to the isomorphisms between two graphs. The nonnegative real solutions to this system are called fractional isomorphisms, and two graphs are fractionally isomorphic if and only if the color refinement algorithm cannot distinguish them (Tinhofer 1986, 1991). We show that, if we drop the nonnegativity constraints, that is, if we look for arbitrary real solutions, then a solution to the linear system exists if and only if, for all t, the two graphs have the same number of length-t walks. We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width k, the k-dimensional Weisfeiler-Leman algorithm, and the level-k Sherali-Adams relaxation of our linear program. We also obtain a partial result for graphs of bounded path width and solutions to our system where we drop the nonnegativity constraints. A consequence of our results is a quasi-linear time algorithm to decide whether, for two given graphs G and H, there is a tree T with Hom(T,G) = Hom(T,H).

cs.DS↗

arXiv2018

An improved isomorphism test for bounded-tree-width graphs

We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time $2^{\mathcal{O}(k^5\log k)}\operatorname{poly} (n)$. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width $k$. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai's algorithm as a black box in one place. We also give a second algorithm which, at the price of a slightly worse running time $2^{\mathcal{O}(k^2 \log k)}\operatorname{poly} (n)$, avoids the use of Babai's algorithm and, more importantly, has the additional benefit that it can also used as a canonization algorithm.

cs.DS↗

arXiv2018

Graph Similarity and Approximate Isomorphism

The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs $G,H$ of the same order $n$ with adjacency matrices $A_G,A_H$, a well-studied measure of similarity is the Frobenius distance \[ \mathrm{dist}(G,H):=\min_π\|A_G^π-A_H\|_F, \] where $π$ ranges over all permutations of the vertex set of $G$, where $A_G^π$ denotes the matrix obtained from $A_G$ by permuting rows and columns according to $π$, and where $\|M\|_F$ is the Frobenius norm of a matrix $M$. The (weighted) graph similarity problem, denoted by SIM (WSIM), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases. It is known that SIM (WSIM) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSIM is best done in the framework of linear algebra. We show that WSIM is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where one of the matrices has a bounded clustering number, a parameter arising from spectral graph drawing techniques.

cs.DS↗