arXiv subjects

Marek Černý

Publications and source records attributed to Marek Černý.

4 recordsLinked to original sources

arXiv2026

Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph $F \in \mathcal{F}$. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems $\textrm{HomInd}(\mathcal{F})$ of deciding homomorphism indistinguishability over $\mathcal{F}$ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of $\textrm{HomInd}(\mathcal{F})$, Seppelt (MFCS 2024) showed that $\textrm{HomInd}(\mathcal{F})$ is in randomised polynomial time for every graph class $\mathcal{F}$ of bounded treewidth that can be defined in counting monadic second-order logic $\mathsf{CMSO}_2$. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in $\mathsf{PTIME}$. For $\mathsf{CMSO}_2$-definable graph classes $\mathcal{F}$ of bounded pathwidth, we improve the previous complexity upper bound for $\textrm{HomInd}(\mathcal{F})$ from $\mathsf{PTIME}$ to $\mathsf{C}_=\mathsf{L}$ and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as $\mathsf{C}_=\mathsf{L}$-complete.

cs.CC↗

arXiv2026

On Numbers of Simplicial Walks and Equivalent Canonizations for Graph Recognition

Two graphs are isomorphic exactly when they admit the same number of homomorphisms from every graph. Hence, a graph is recognized up to isomorphism by homomorphism counts over the class of all graphs. Restricting to a specific graph class yields some natural isomorphism relaxations and modulates recognition to particular graph properties. A notable restriction is to the classes of bounded treewidth, yielding the isomorphism relaxation of Weisfeiler--Leman refinement (WL), as shown by Dvořák [JGT 2010]. The properties recognized by WL are exactly those definable in fragments of first-order logic with counting quantifiers, as shown by Cai, Fürer, and Immerman [Comb. 1992]. We characterize the restriction to the classes of bounded pathwidth by numbers of simplicial walks, and formalize it into a refinement procedure (SW). The properties recognized by SW are exactly those definable in fragments of restricted-conjunction first-order logic with counting quantifiers, introduced by Montacute and Shah [LMCS 2024]. Unlike WL, computing SW directly is not polynomial-time in general. We address this by representing SW in terms of multiplicity automata. We equip these automata with an involution, simplifying the canonization to standard forward reduction and omitting the backward one. The resulting canonical form is computable in time $O(kn^{3k})$ for any graph on $n$ vertices and the restriction to pathwidth at most $k$.

cs.DM↗

arXiv2025

Caterpillar GNN: Replacing Message Passing with Efficient Aggregation

Message-passing graph neural networks (MPGNNs) dominate modern graph learning. Typical efforts enhance MPGNN's expressive power by enriching the adjacency-based aggregation. In contrast, we introduce an efficient aggregation over walk incidence-based matrices that are constructed to deliberately trade off some expressivity for stronger and more structured inductive bias. Our approach allows for seamless scaling between classical message-passing and simpler methods based on walks. We rigorously characterize the expressive power at each intermediate step using homomorphism counts over a hierarchy of generalized caterpillar graphs. Based on this foundation, we propose Caterpillar GNNs, whose robust graph-level aggregation successfully tackles a benchmark specifically designed to challenge MPGNNs. Moreover, we demonstrate that, on real-world datasets, Caterpillar GNNs achieve comparable predictive performance while significantly reducing the number of nodes in the hidden layers of the computational graph.

cs.LG↗

arXiv2021

ICLR 2021 Challenge for Computational Geometry & Topology: Design and Results

This paper presents the computational challenge on differential geometry and topology that happened within the ICLR 2021 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide creative contributions to the fields of computational geometry and topology through the open-source repositories Geomstats and Giotto-TDA. The challenge attracted 16 teams in its two month duration. This paper describes the design of the challenge and summarizes its main findings.

cs.CG↗