[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81917-en":3,"doc-seo-81917-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81917,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","The Fine-Grained Complexity of Counting Hypergraph Motifs","Introduced by Lee, Ko, and Shin, hypergraph motifs are connected subhypergraphs whose three hyperedges intersect according to a Venn-diagram pattern. Prior work achieved cubic-time exact and approximate counting in the number of hyperedges, even for bounded rank. This paper delivers a complete fine-grained view of parameterized exact motif counting by rank: every Venn diagram admits fixed-parameter near-quadratic time, while fixed-parameter near-linear time occurs exactly for degenerate diagrams. Nondegenerate cases require failure of either the Triangle Hypothesis or the Hyperclique Hypothesis. The work extends to generalized motifs and develops reductions via hypergraph homomorphism counts.","arXiv :2607 .05040v 1 [ cs .CC] 6 Jul 2026  \nThe Fine-Grained Complexity of Counting Hypergraph Motifs  \nMadhumitha Krishnakumar 1 and Marc Roth 1  \n1 School of Electronic Engineering & Computer Science, Queen Mary University of London  \nJuly 7, 2026  \nAbstract  \nIntroduced by Lee, Ko, and Shin (VLDB 2020), a hypergraph motif is a connected subhypergraph consisting of three hyperedges whose intersections satisfy a prescribed pattern. Such patterns are represented by Venn diagrams V ∈ {0, 1}7 , indicating which of the seven regions determined by three sets must be empty or non-empty. Lee et al. designed and implemented exact and approximate algorithms for counting, in a hypergraph G, the motifs specified by V; their algorithms run in worst-case cubic time in the number of hyperedges of G. This cubic worst case can occur even for hypergraphs of bounded rank, and already for 2-uniform hypergraphs, that is, for simple graphs.  \nIn this work, we give a complete fine-grained picture of the parameterised com˜ plexity of exact hyper  \ngraph motif counting with respect to the rank of the input hypergraph. We use O to hide polylogarithmic factors in the input size. First, we show that every Venn diagram V admits an exact counting algorithm running in FPT-near-quadratic time,  \nf (rank(G)) · ˜O(|E(G)| 2 ) ,  \nfor some computable function f. Second, we precisely characterise when this can be improved to FPTnear-linear time. We prove that such an algorithm exists exactly for the degenerate Venn diagrams, namely those that force one of the three hyperedges to be fully contained in another. For all nondegenerate Venn diagrams, we show that no FPT-near-linear-time algorithm exists unless either the Triangle Hypothesis or the Hyperclique Hypothesis fails. Exact hypergraph motif counting is thus always fixed-parameter near-quadratic in the rank, and the degenerate Venn diagrams are precisely the cases admitting fixed-parameter near-linear time. In particular, for bounded-rank hypergraphs, where f (rank(G)) = O(1), our results give near-quadratic upper bounds in general and a tight classification of the near-linear-time cases.  \nWe also study generalised hypergraph motifs defined by emptiness constraints on all intersections of khyperedges. We establish new parameterised upper and lower bounds when parameterised by k+rank(G) . We further argue that obtaining matching bounds is closely tied to the hypergraph homomorphism counting problem, whose complexity remains open for unbounded-rank hypergraphs.  \nOur results build on the recently established hypergraph homomorphism basis of Bressan, Brinkmann, Dell, Roth, and Wellnitz (SODA 2026) . The main technical challenge is the proof of the fine-grained lower bound which requires us to express the number of hypergraph motifs as a linear combination of homomorphism counts and to show that cyclic terms survive with non-zero coefficients for all nondegenerate Venn diagrams. This task is complicated significantly by the fact that multiple non-isomorphic subhypergraphs can satisfy the same Venn diagram. We overcome this challenge by introducing and analysing an intermediate coloured version of the problem.  \nFigure 1: (Left:) An indexing of the intersections of three sets. (Centre): Illustration of the Venn diagram V = (0 , 1 , 1 , 1 , 1 , 1 , 1) . (Right): A hypergraph satisfying V.  \n1 Introduction  \nMotif counting refers to the problem of computing the number of occurrences of a small pattern in a large host network. Examples include subgraph counting, induced subgraph counting, as well as counting answers toa query in a relational database. A 2002 landmark result of Milo, Shen-Orr, Itzkovitz, Kashtan, Chklovskii, and Alon [44] identified a variety of subgraph patterns, subsequently called network motifs, that appear with significantly higher frequency in real-life networks than their expected frequency in random networks, and that increased frequencies of certain patterns correlate with global features ","cbCaijNtewmEnBlS","https://ap.wps.com/l/cbCaijNtewmEnBlS","pdf",964037,7,1,36,"English","en",105,"# Abstract\n# Introduction\n## Motif counting background\n## Hypergraph motifs and Venn-diagram patterns","[{\"question\":\"What is a hypergraph motif in this paper’s setting?\",\"answer\":\"A hypergraph motif is a connected subhypergraph formed by three hyperedges whose pairwise/three-way intersections follow a prescribed Venn-diagram emptiness/non-emptiness pattern.\"},{\"question\":\"What running times are established for exact hypergraph motif counting?\",\"answer\":\"For every Venn diagram pattern, exact counting has fixed-parameter near-quadratic time in the rank parameter. Fixed-parameter near-linear time exists exactly for degenerate Venn diagrams, where one hyperedge is fully contained in another.\"},{\"question\":\"How does the paper relate nondegenerate cases to major conjectures?\",\"answer\":\"For nondegenerate Venn diagrams, no fixed-parameter near-linear-time algorithm exists unless either the Triangle Hypothesis or the Hyperclique Hypothesis fails.\"}]","The Fine-Grained Complexity of Counting Hypergraph Motifs | PDF",1784177041,91,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"the-fine-grained-complexity-of-counting-hypergraph-motifs","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/the-fine-grained-complexity-of-counting-hypergraph-motifs/81917/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What is a hypergraph motif in this paper’s setting?","Question",{"text":77,"@type":78},"A hypergraph motif is a connected subhypergraph formed by three hyperedges whose pairwise/three-way intersections follow a prescribed Venn-diagram emptiness/non-emptiness pattern.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"What running times are established for exact hypergraph motif counting?",{"text":82,"@type":78},"For every Venn diagram pattern, exact counting has fixed-parameter near-quadratic time in the rank parameter. Fixed-parameter near-linear time exists exactly for degenerate Venn diagrams, where one hyperedge is fully contained in another.",{"name":84,"@type":75,"acceptedAnswer":85},"How does the paper relate nondegenerate cases to major conjectures?",{"text":86,"@type":78},"For nondegenerate Venn diagrams, no fixed-parameter near-linear-time algorithm exists unless either the Triangle Hypothesis or the Hyperclique Hypothesis fails.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,117,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":113,"doc_module":4,"doc_module_name":47,"category_name":114,"show_sort_weight":115,"slug":116},6,"Technology",50,"technology",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":108,"slug":139},19,"General","general"]