[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84289-en":3,"doc-seo-84289-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84289,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Homomorphism Indistinguishability Beyond Graphs: Relational Weisfeiler–Leman and Hypertree Width","The document studies the Weisfeiler–Leman (WL) family of refinement algorithms for graph isomorphism and extends the expressiveness analysis to relational structures of higher arity. It surveys prior results linking k-dimensional WL indistinguishability to homomorphism counts over bounded (generalised) hypertree width. The work resolves an open problem by defining a k-dimensional variant of relational colour refinement (k-RCR) and proving an equivalence with indistinguishability by homomorphism numbers from bounded hypertree-width structures. It further introduces fractional k-RCR and a new algorithm, k-HyperOWL, operating directly on relational inputs and enabling constructive preprocessing for homomorphism counting.","arXiv :2607 .07934v 1 [ cs .DS] 8 Jul 2026  \nHomomorphism Indistinguishability Beyond Graphs: Relational Weisfeiler–Leman and Hypertree Width  \nPanagiotis Aivasiliotis \\# Hasso Plattner Institute, University of Potsdam  \nAndreas Gbel \\#  \nHasso Plattner Institute, University of Potsdam  \nMatthias Lanzinger \\# Institute of Logic and Computation, TU Wien  \nMarc Roth \\#  \nSchool of Electronic Engineering and Computer Science, Queen Mary University of London  \n~~ Abstract ~~  \nThe Weisfeiler-Leman (WL) algorithm is one of the most influential heuristics for the graph isomorphism problem and constitutes a cornerstone of Babai’s celebrated quasi-polynomial-time isomorphism test [STOC’16] . Starting with the seminal work of Cai, Frer, and Immerman [Combinatorica’92] the expressive power of WL has been extensively studied over the past 35 years in the contexts of descriptive complexity, logics, graph neural networks, and the theory of homomorphism indistinguishabily, the latter of which dates back to early works of Lovsz. In a landmark result, Grohe, Dell, and Rattan [ICALP’18] proved that two graphs are indistinguishable by the 􀀺-dimensional WL algorithm if and only if they are indistinguishable by homomorphism counts from graphs of treewidth at most 􀀺 .  \nAn intrinsic question is whether there is a natural version of the WL algorithm which operates on hypergraphsand relational structures of higher arity. Scheidt [ICALP’24] argues that a “proper” version of a 􀀺-dimensional relational WL should admit an equivalent characterisation via homomorphism indistinguishability along bounded generalised hypertree width (GHW) and poses the search for such a version for all 􀀺 ≥ 1 as an open problem. In follow-up work, Scheidt and Schweikardt [MFCS’25] confirmed this for 􀀺 = 1 by defining relational colour refinement (RCR), a 1-dimensional WL variant, and by showing the respective homomorphism-indistinguishability result from (􀀍-)acyclic structures (i.e. structures with GHW=1) . Nevertheless, the question remains open for all 􀀺 > 1.  \nIn this work, we provide a definitive and affirmative resolution: we develop a 􀀺-dimensional version of RCR (􀀺-RCR for short) and show that two structures A and B are insdistinguishable by 􀀺-RCR if and only if they have the same number of homomorphisms from all structures C of generalised hypertreewidth at most 􀀺 . Moreover, we introduce a more intricate “fractional” version of 􀀺-RCR and show that A and B are insdistinguishable by fractional 􀀺-RCR if and only if they have the same number of homomorphisms from all structures C of a variant of fractional hypertreewidth at most 􀀺 .  \nLast but not least, we develop “􀀺-HyperOWL”, the first relational 􀀺-WL algorithm that, in contrast to existing attempts for relational WL (including 􀀺-RCR), operates directly on the given relational structure rather than relying on a transformation to a graph-like structure of rank at most 2. We show that 􀀺-HyperOWL is at least as expressive as 􀀺-RCR and that, given a structure A, 􀀺-HyperOWL can compute 􀁃 iterative refinements in time 􀀤(􀁃 · |A | 􀀺+1) where the hidden constant only depends on 􀀺, and the signature and rank of A. Moreover, we are able to use 􀀺-HyperOWL as a constructive preprocessing routine to design an algorithm for counting homomorphisms from structures of generalised hypertreewidth at most 􀀺 operating on the colouring produced by 􀀺-HyperOWL. This algorithm constitutes a direct generalisation of the recent result of Lanzinger and Barcel [ICLR 2024] from structures of arity 2 to arbitrary structures.  \nFunding Andreas Gbel: Postdoc Network Brandenburg  \n 1  Extended Abstract  \nThe colour refinement algorithm is a famous heuristic for the graph isomorphism problem and its inception dates back to 1965 when Morgan introduced an iterative neighbourhood-refinement method for encoding chemical structures [35] . In its modern form, the algorithm tests, in near linear time [6], whether two graphs are isomorphic, or simply put, wheth","cbCaidbRbc1vqiFT","https://ap.wps.com/l/cbCaidbRbc1vqiFT","pdf",607191,4,1,52,"English","en",105,"# Abstract\n# Colour refinement and k-WL background\n## Iterative neighbourhood refinement\n## Vertex-to-tuple lifting and expressiveness","[{\"question\":\"What is the main problem addressed regarding WL algorithms?\",\"answer\":\"The document asks whether a natural higher-arity version of Weisfeiler–Leman can be formulated for relational structures and characterised via homomorphism indistinguishability over bounded hypertree width.\"},{\"question\":\"What does the paper prove about k-RCR and homomorphism counts?\",\"answer\":\"It defines a k-dimensional relational colour refinement (k-RCR) and shows that two structures are indistinguishable by k-RCR exactly when they have the same numbers of homomorphisms from all structures with generalised hypertree width at most k.\"},{\"question\":\"How is k-HyperOWL different from existing relational WL attempts?\",\"answer\":\"k-HyperOWL is designed to operate directly on the given relational structure rather than transforming it into a graph-like structure of bounded rank, while matching or exceeding the expressiveness of k-RCR and supporting homomorphism-counting 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is the main problem addressed regarding WL algorithms?","Question",{"text":75,"@type":76},"The document asks whether a natural higher-arity version of Weisfeiler–Leman can be formulated for relational structures and characterised via homomorphism indistinguishability over bounded hypertree width.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the paper prove about k-RCR and homomorphism counts?",{"text":80,"@type":76},"It defines a k-dimensional relational colour refinement (k-RCR) and shows that two structures are indistinguishable by k-RCR exactly when they have the same numbers of homomorphisms from all structures with generalised hypertree width at most k.",{"name":82,"@type":73,"acceptedAnswer":83},"How is k-HyperOWL different from existing relational WL attempts?",{"text":84,"@type":76},"k-HyperOWL is designed to operate directly on the given relational structure rather than transforming it into a graph-like structure of bounded rank, while matching or exceeding 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