arXiv subjects

Matthias Lanzinger

Publications and source records attributed to Matthias Lanzinger.

At least 19 recordsLinked to original sources

arXiv2026

The Boolean Power of ReLU

We prove that, on finite simple undirected graphs equipped with a single Boolean node feature, the Boolean queries expressible in $\Sigma$-MPLang, for any collection $\Sigma$ of eventually constant activation functions and with arbitrary real coefficients, form a strict subclass of the Boolean queries expressible in ReLU-MPLang. We thereby settle a recently posed open problem: whether ReLU-MPLang is more powerful than trReLU-MPLang when it comes to Boolean queries. In particular, this implies that ReLU-GNNs are strictly more expressive than {TrReLU,id}-GNNs with respect to Boolean queries on Boolean-featured graphs.

cs.LG↗

arXiv2026

Homomorphism Indistinguishability Beyond Graphs: Relational Weisfeiler--Leman and Hypertree Width

The Weisfeiler--Leman (WL) algorithm is one of the most influential heuristics for the graph isomorphism problem. The expressive power of WL has been extensively studied in the contexts of descriptive complexity, logics, graph neural networks, and the theory of homomorphism indistinguishabily. Notably, two graphs are indistinguishable by the $k$-dimensional WL algorithm if and only if they are indistinguishable by homomorphism-counts from graphs of treewidth at most $k$. An intrinsic question is to find a natural version of the WL algorithm for relational structures of higher arity admitting an equivalent characterisation via homomorphism indistinguishability along bounded generalised hypertree width (GHW). Scheidt and Schweikardt solved this for $k=1$ by defining the RCR algorithm and showing indistinguishability from $\alpha$-acyclic structures. In this work, we resolve this for all $k\ge1$: we develop $k$-RCR and show that two structures $\mathcal{A}$ and $\mathcal{B}$ are insdistinguishable by $k$-RCR if and only if they have the same homomorphism-counts from all structures $\mathcal{C}$ of generalised hypertreewidth $\le k$. Moreover, we introduce a ``fractional'' version of $k$-RCR and show that two structures are insdistinguishable by fractional $k$-RCR if and only if they have the same homomorphism-counts from all structures with (a variant of) fractional hypertreewidth at most $k$. Last, we develop $k$-HyperOWL, the first relational WL algorithm operating directly on a relational structure. We show that $k$-HyperOWL is as expressive as $k$-RCR and that, given a structure $\mathcal{A}$, $k$-HyperOWL can compute $t$ iterative refinements in time $O(t|\mathcal{A}|^{k+1})$. Moreover, the colouring produced by $k$-HyperOWL can be used as a constructive preprocessing routine for counting homomorphisms from structures of generalised hypertreewidth $\le k$.

cs.DS↗

arXiv2026

Cuts and Gauges for Submodular Width

Submodular width is a central structural measure governing the complexity of conjunctive query evaluation. In this paper we recast submodular width in geometric terms. We how that submodular width can be approximated, up to a factor $3/2$, by a new branchwidth parameter defined in terms of edge separations in the hypergraph and the costs induced on them by admissible submodular functions. This reformulation turns lower bounds on submodular width into the problem of constructing well-balanced edge separations whose induced cost remains small. We then express this connection through a variational characterisation in terms of a convex body. Using these tools, we relate submodular width to more familiar graph-theoretic notions, including line-graph treewidth and multicommodity flow, and obtain general conditions under which submodular width is tightly linked to generalised hypertree width. In particular, under various natural conditions we show that \[ subw(H) \in \Omega \left(\frac{ghw(H)}{\log ghw(H)} \right). \]

cs.DS↗

arXiv2025

A Logical View of GNN-Style Computation and the Role of Activation Functions

We study the numerical and Boolean expressiveness of MPLang, a declarative language that captures the computation of graph neural networks (GNNs) through linear message passing and activation functions. We begin with A-MPLang, the fragment without activation functions, and give a characterization of its expressive power in terms of walk-summed features. For bounded activation functions, we show that (under mild conditions) all eventually constant activations yield the same expressive power - numerical and Boolean - and that it subsumes previously established logics for GNNs with eventually constant activation functions but without linear layers. Finally, we prove the first expressive separation between unbounded and bounded activations in the presence of linear layers: MPLang with ReLU is strictly more powerful for numerical queries than MPLang with eventually constant activation functions, e.g., truncated ReLU. This hinges on subtle interactions between linear aggregation and eventually constant non-linearities, and it establishes that GNNs using ReLU are more expressive than those restricted to eventually constant activations and linear layers.

cs.LG↗

arXiv2025

From Alternation to FPRAS: Toward a Complexity Classification of Approximate Counting

Counting problems are fundamental across mathematics and computer science. Among the most subtle are those whose associated decision problem is solvable in polynomial time, yet whose exact counting version appears intractable. For some such problems, however, one can still obtain efficient randomized approximation in the form of a fully polynomial randomized approximation scheme (FPRAS). Existing proofs of FPRAS existence are often highly technical and problem-specific, offering limited insight into a more systematic complexity-theoretic account of approximability. In this work, we propose a machine-based framework for establishing the existence of an FPRAS beyond previous uniform criteria. Our starting point is alternating computation: we introduce a counting model obtained by equipping alternating Turing machines with a transducer-style output mechanism, and we use it to define a corresponding counting class spanALP. We show that every problem in spanALP admits an FPRAS, yielding a reusable sufficient condition that can be applied via reductions to alternating logspace, polynomial-time computation with output. We situate spanALP in the counting complexity landscape as strictly between #L and TotP (assuming RP $\neq$ NP) and observe interesting conceptual and technical gaps in the current machinery counting complexity. Moreover, as an illustrative application, we obtain an FPRAS for counting answers to counting the answers Dyck-constrained path queries in edge-labeled graphs, i.e., counting the number of distinct labelings realized by s-t walks whose label sequence is well-formed with respect to a Dyck-like language. To our knowledge, no FPRAS was previously known for this setting. We expect the alternating-transducer characterization to provide a broadly applicable tool for establishing FPRAS existence for further counting problems.

cs.CC↗

arXiv2025

Message Passing on the Edge: Towards Scalable and Expressive GNNs

Graph neural networks (GNNs) are widely used in graph learning and most architectures propagate information by passing messages between vertices. In this work, we shift our attention to GNNs that perform message passing on edges and introduce EB-1WL, an edge-based color-refinement test, and a corresponding architecture, EB-GNN. Our EB-GNN architecture is inspired by the classic triangle-counting algorithm of Chiba and Nishizeki and passes messages along edges and triangles. Our contributions are as follows: (1) Theoretically, we show that EB-1WL is significantly more expressive than 1WL. We provide a complete logical characterization of EB-1WL in first-order logic, along with distinguishability results via homomorphism counting. To the best of our knowledge, EB-GNN has the strongest theoretical expressivity guarantees among edge-based message-passing GNNs in the literature. (2) Unlike many GNN architectures that are more expressive than 1WL, we prove that EB-1WL and EB-GNN admit near-linear time and memory usage on practical graph learning workloads. (3) We show in experiments that EB-GNN is a highly efficient general-purpose architecture: it substantially outperforms simple MPNNs and remains competitive with task-specialized state-of-the-art GNNs at substantially lower computational cost.

cs.LG↗

arXiv2025

FPT Parameterisations of Fractional and Generalised Hypertree Width

We present the first fixed-parameter tractable (FPT) algorithms for exact computation of generalized hypertree width (ghw) and fractional hypertree width (fhw). Our algorithms are parameterized by the target width, the rank, and the maximum degree of the input hypergraph. More generally, we show that testing f-width is in FPT for a broad class of width functions that we call manageable. This class contains the edge cover number $\rho$ and its fractional relaxation $\rho^*$, and thus covers both generalized and fractional hypertree width. We additionally extend our framework to also obtain an fpt algorithm for computing a discretized version of adaptive width. Our approach extends a recent algorithm for treewidth (Boja\'ncyk & Pilipczuk, LMCS 2022) that utilises monadic second-order transductions. To extend this idea beyond treewidth we develop new combinatorial machinery around elimination forests in hypergraphs, culminating in a structural normal form for optimal witnesses that makes transduction-based optimisation applicable in the much more general context of manageable width functions. This yields the first exact FPT algorithms for these measures under any nontrivial parameterisation and provides structural tools that may enable more direct optimisation algorithms

cs.DS↗

arXiv2025

Selective Use of Yannakakis' Algorithm to Improve Query Performance: Machine Learning to the Rescue

Query optimization has played a central role in database research for decades. However, more often than not, the proposed optimization techniques lead to a performance improvement in some, but not in all, situations. Therefore, we urgently need a methodology for designing a decision procedure that decides for a given query whether the optimization technique should be applied or not. In this work, we propose such a methodology with a focus on Yannakakis-style query evaluation as our optimization technique of interest. More specifically, we formulate this decision problem as an algorithm selection problem and we present a Machine Learning based approach for its solution. Empirical results with several benchmarks on a variety of database systems show that our approach indeed leads to a statistically significant performance improvement.

cs.DB↗

arXiv2024

Soft and Constrained Hypertree Width

Hypertree decompositions provide a way to evaluate Conjunctive Queries (CQs) in polynomial time, where the exponent of this polynomial is determined by the width of the decomposition. In theory, the goal of efficient CQ evaluation therefore has to be a minimisation of the width. However, in practical settings, it turns out that there are also other properties of a decomposition that influence the performance of query evaluation. It is therefore of interest to restrict the computation of decompositions by constraints and to guide this computation by preferences. To this end, we propose a novel framework based on candidate tree decompositions, which allows us to introduce soft hypertree width (shw). This width measure is a relaxation of hypertree width (hw); it is never greater than hw and, in some cases, shw may actually be lower than hw. Most importantly, shw preserves the tractability of deciding if a given CQ is below some fixed bound, while offering more algorithmic flexibility. In particular, it provides a natural way to incorporate preferences and constraints into the computation of decompositions. A prototype implementation and preliminary experiments confirm that this novel framework can indeed have a practical impact on query evaluation.

cs.DB↗

arXiv2024

Homomorphism Counts as Structural Encodings for Graph Learning

Graph Transformers are popular neural networks that extend the well-known Transformer architecture to the graph domain. These architectures operate by applying self-attention on graph nodes and incorporating graph structure through the use of positional encodings (e.g., Laplacian positional encoding) or structural encodings (e.g., random-walk structural encoding). The quality of such encodings is critical, since they provide the necessary $\textit{graph inductive biases}$ to condition the model on graph structure. In this work, we propose $\textit{motif structural encoding}$ (MoSE) as a flexible and powerful structural encoding framework based on counting graph homomorphisms. Theoretically, we compare the expressive power of MoSE to random-walk structural encoding and relate both encodings to the expressive power of standard message passing neural networks. Empirically, we observe that MoSE outperforms other well-known positional and structural encodings across a range of architectures, and it achieves state-of-the-art performance on a widely studied molecular property prediction dataset.

cs.LG↗

arXiv2024

Avoiding Materialisation for Guarded Aggregate Queries

Optimising queries with many joins is known to be a hard problem. The explosion of intermediate results as opposed to a much smaller final result poses a serious challenge to modern database management systems (DBMSs). This is particularly glaring in case of analytical queries that join many tables, but ultimately only output comparatively small aggregate information. Analogous problems are faced by graph database systems when processing analytical queries with aggregates on top of complex path queries. In this work, we propose novel optimisation techniques both, on the logical and physical level, that allow us to avoid the materialisation of join results for certain types of aggregate queries. The key to these optimisations is the notion of guardedness, by which we impose restrictions on the occurrence of attributes in GROUP BY clauses and in aggregate expressions. The efficacy of our optimisations is validated through their implementation in Spark SQL and extensive empirical evaluation on various standard benchmarks.

cs.DB↗

arXiv2024

Fuzzy Datalog$^\exists$ over Arbitrary t-Norms

One of the main challenges in the area of Neuro-Symbolic AI is to perform logical reasoning in the presence of both neural and symbolic data. This requires combining heterogeneous data sources such as knowledge graphs, neural model predictions, structured databases, crowd-sourced data, and many more. To allow for such reasoning, we generalise the standard rule-based language Datalog with existential rules (commonly referred to as tuple-generating dependencies) to the fuzzy setting, by allowing for arbitrary t-norms in the place of classical conjunctions in rule bodies. The resulting formalism allows us to perform reasoning about data associated with degrees of uncertainty while preserving computational complexity results and the applicability of reasoning techniques established for the standard Datalog setting. In particular, we provide fuzzy extensions of Datalog chases which produce fuzzy universal models and we exploit them to show that in important fragments of the language, reasoning has the same complexity as in the classical setting.

cs.AI↗

arXiv2024

Homomorphism Counts for Graph Neural Networks: All About That Basis

A large body of work has investigated the properties of graph neural networks and identified several limitations, particularly pertaining to their expressive power. Their inability to count certain patterns (e.g., cycles) in a graph lies at the heart of such limitations, since many functions to be learned rely on the ability of counting such patterns. Two prominent paradigms aim to address this limitation by enriching the graph features with subgraph or homomorphism pattern counts. In this work, we show that both of these approaches are sub-optimal in a certain sense and argue for a more fine-grained approach, which incorporates the homomorphism counts of all structures in the ``basis'' of the target pattern. This yields strictly more expressive architectures without incurring any additional overhead in terms of computational complexity compared to existing approaches. We prove a series of theoretical results on node-level and graph-level motif parameters and empirically validate them on standard benchmark datasets.

cs.LG↗

arXiv2023

FPT Approximation of Generalised Hypertree Width for Bounded Intersection Hypergraphs

Generalised hypertree width ($ghw$) is a hypergraph parameter that is central to the tractability of many prominent problems with natural hypergraph structure. Computing $ghw$ of a hypergraph is notoriously hard. The decision version of the problem, checking whether $ghw(H) \leq k$, is paraNP-hard when parameterised by $k$. Furthermore, approximation of $ghw$ is at least as hard as approximation of Set-Cover, which is known to not admit any fpt approximation algorithms. Research in the computation of ghw so far has focused on identifying structural restrictions to hypergraphs -- such as bounds on the size of edge intersections -- that permit XP algorithms for $ghw$. Yet, even under these restrictions that problem has so far evaded any kind of fpt algorithm. In this paper we make the first step towards fpt algorithms for $ghw$ by showing that the parameter can be approximated in fpt time for graphs of bounded edge intersection size. In concrete terms we show that there exists an fpt algorithm, parameterised by $k$ and $d$, that for input hypergraph $H$ with maximal cardinality of edge intersections $d$ and integer $k$ either outputs a tree decomposition with $ghw(H) \leq 4k(k+d+1+)(2k-1)$, or rejects, in which case it is guaranteed that $ghw(H) > k$. Thus, in the special case, of hypergraphs of bounded edge intersection, we obtain an fpt $O(k^3)$-approximation algorithm for $ghw$.

cs.DS↗

arXiv2023

On the Power of the Weisfeiler-Leman Test for Graph Motif Parameters

Seminal research in the field of graph neural networks (GNNs) has revealed a direct correspondence between the expressive capabilities of GNNs and the $k$-dimensional Weisfeiler-Leman ($k$WL) test, a widely-recognized method for verifying graph isomorphism. This connection has reignited interest in comprehending the specific graph properties effectively distinguishable by the $k$WL test. A central focus of research in this field revolves around determining the least dimensionality $k$, for which $k$WL can discern graphs with different number of occurrences of a pattern graph $P$. We refer to such a least $k$ as the WL-dimension of this pattern counting problem. This inquiry traditionally delves into two distinct counting problems related to patterns: subgraph counting and induced subgraph counting. Intriguingly, despite their initial appearance as separate challenges with seemingly divergent approaches, both of these problems are interconnected components of a more comprehensive problem: "graph motif parameters". In this paper, we provide a precise characterization of the WL-dimension of labeled graph motif parameters. As specific instances of this result, we obtain characterizations of the WL-dimension of the subgraph counting and induced subgraph counting problem for every labeled pattern $P$. We additionally demonstrate that in cases where the $k$WL test distinguishes between graphs with varying occurrences of a pattern $P$, the exact number of occurrences of $P$ can be computed uniformly using only local information of the last layer of a corresponding GNN. We finally delve into the challenge of recognizing the WL-dimension of various graph parameters. We give a polynomial time algorithm for determining the WL-dimension of the subgraph counting problem for given pattern $P$, answering an open question from previous work.

cs.LG↗

arXiv2023

Structure-Guided Query Evaluation: Towards Bridging the Gap from Theory to Practice

Join queries involving many relations pose a severe challenge to today's query optimisation techniques. To some extent, this is due to the fact that these techniques do not pay sufficient attention to structural properties of the query. In stark contrast, the Database Theory community has intensively studied structural properties of queries (such as acyclicity and various notions of width) and proposed efficient query evaluation techniques through variants of Yannakakis' algorithm. However, although most queries in practice actually are acyclic or have low width, structure-guided query evaluation techniques based on Yannakakis' algorithm have not found their way into mainstream database technology yet. The goal of this work is to address this gap between theory and practice and to demonstrate that the consideration of query structure can improve query evaluation performance on modern DBMSs significantly in cases that have been traditionally challenging. In particular, we study the performance of structure-guided query evaluation in three architecturally distinct DBMSs by rewriting SQL queries into a sequence of SQL statements that express an execution of Yannakakis' algorithm. Moreover, we identify a class of queries that is particularly well suited for our approach and allows query answering in a variety of common scenarios without materializing any join. Through empirical evaluation we show that structure-guided query evaluation can make the evaluation of many difficult join queries feasible whereas their evaluation requires a prohibitive amount of time and memory on current DBMSs.

cs.DB↗

arXiv2022

The Complexity of Pattern Counting in Directed Graphs, Parameterised by the Outdegree

We study the fixed-parameter tractability of the following fundamental problem: given two directed graphs $\vec H$ and $\vec G$, count the number of copies of $\vec H$ in $\vec G$. The standard setting, where the tractability is well understood, uses only $|\vec H|$ as a parameter. In this paper we take a step forward, and adopt as a parameter $|\vec H|+d(\vec G)$, where $d(\vec G)$ is the maximum outdegree of $|\vec G|$. Under this parameterization, we completely characterize the fixed-parameter tractability of the problem in both its non-induced and induced versions through two novel structural parameters, the fractional cover number $\rho^*$ and the source number $\alpha_s$. On the one hand we give algorithms with running time $f(|\vec H|,d(\vec G)) \cdot |\vec G|^{\rho^*\!(\vec H)+O(1)}$ and $f(|\vec H|,d(\vec G)) \cdot |\vec G|^{\alpha_s(\vec H)+O(1)}$ for counting respectively the copies and induced copies of $\vec H$ in $\vec G$; on the other hand we show that, unless the Exponential Time Hypothesis fails, for any class $\vec C$ of directed graphs the (induced) counting problem is fixed-parameter tractable if and only if $\rho^*(\vec C)$ ($\alpha_s(\vec C)$) is bounded. These results explain how the orientation of the pattern can make counting easy or hard, and prove that a classic algorithm by Chiba and Nishizeki and its extensions (Chiba, Nishizeki SICOMP 85; Bressan Algorithmica 21) are optimal unless ETH fails.

cs.CC↗

arXiv2022

Incremental Updates of Generalized Hypertree Decompositions

Structural decomposition methods, such as generalized hypertree decompositions, have been successfully used for solving constraint satisfaction problems (CSPs). As decompositions can be reused to solve CSPs with the same constraint scopes, investing resources in computing good decompositions is beneficial, even though the computation itself is hard. Unfortunately, current methods need to compute a completely new decomposition even if the scopes change only slightly. In this paper, we make the first steps toward solving the problem of updating the decomposition of a CSP $P$ so that it becomes a valid decomposition of a new CSP $P'$ produced by some modification of $P$. Even though the problem is hard in theory, we propose and implement a framework for effectively updating GHDs. The experimental evaluation of our algorithm strongly suggests practical applicability.

cs.AI↗