[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82595-en":3,"doc-seo-82595-105":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},82595,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Query Complexity of Hypergraph Connectivity and Learnability using CUT Oracles","The paper studies CUT queries as a way to uncover the structure of unknown hypergraphs, focusing on how much connectivity information can be identified when exact edge learning is generally impossible due to identifiability barriers. It presents a zero-error randomized O(n) expected-query algorithm to determine connected components of weighted hypergraphs via independent families and auxiliary graph connectivity techniques. It further characterizes exact learnability by hyperedge parity, giving reconstruction and certificate algorithms for even hypergraphs and improved k-connectivity complexity for linear cases.","arXiv :2607 .0 12 16v 1 [ cs .DS] 1 Jul 2026  \nQuery Complexity of Hypergraph Connectivity and Learnability  \nusing CUT Oracles  \nDeeparnab Chakrabarty* Hang Liao†  \nAbstract  \nWe investigate the power of CUT queries to reveal the structure of unknown hypergraphs. While simple graphs allow for optimal O (n)-query connectivity algorithms, hypergraphs face a fundamental identifiability barrier in that distinct hypergraphs can share identical cut-profiles, making exact edge learning impossible in general, a primitive crucial in the graph connectivity algorithms.  \nWe first present a zero-error randomized algorithm that identifies the connected components of any weighted hypergraph using O (n) expected queries, matching the Ω(n) lower bound. This approach bypasses the reconstruction barrier by introducing the notion of “independent families”—vertex subpartitions that do not share hyperedges—and iteratively coarsening them using auxiliary weighted graph connectivity techniques [Liao-Chakrabarty, 2024] .  \nSecond, we demonstrate that the impossibility of exact learning depends on hyperedge parity. For even-parity hypergraphs, we show that the structure is reconstructible using a Mbius transform on the CUT function to implement binary-search-style vertex identification. This yields d˜eterministic algorithms for obtaining k-connectivity certificates for r-bounded even hypergraphs in Or (kn)˜queries. Finally, we bypass parity and rank constraints for linear hypergraphs, achieving a sub˜ quadratic O (kn1.5 )  \nquery complexity for k-connectivity. This significantly improves upon the general O(n2 ) bound derived via symmetric submodular function minimization.  \n* Dartmouth College, Email: [deeparnab@dartmouth.edu](deeparnab@dartmouth.edu)  \n†Palo Alto Networks, Email: [hangliao98@gmail.com](hangliao98@gmail.com. Work done)[. Work done](hangliao98@gmail.com. Work done) as a graduate student at Dartmouth College.  \n1 Introduction  \nMotivated by the complexity of symmetric submodular function minimization (SFM), Rubinstein, Schramm, and Weinberg [34] introduced the CUT-query model to study the connectivity of an undirected graph G =(V, E) whose vertex set on n vertices is known but the edge set is unknown. For any subset S ⊆ V , CUT (S) returns the number/weight of edges crossing from S to V \\ S. A simple binary-search style idea allows one to sample a random edge incident on a vertex in O(log n)-many queries. This simple but crucial primitive, along with many other ideas, has been key to many recent results [30, 4, 27, 5, 3, 7, 29, 1, 22, 23, 24]; for instance, this has culminated in the recent zero-error randomized algorithms [3, 29] that determine the connected components of a (possibly weighted) graph making O (n) queries in expectation, and this is tight [5] .  \nIn this paper, we initiate the systematic study of hypergraphs in the CUT-query model: CUT (S) now returns the number/weight of hyperedges e ∈ E which intersect both S and V \\ S. While this function still emmuhescefmthra,rindsuitistibmon fnctulmpargergrfunaphraphcsoochn– iyphyirapidnghenst(2aT) qfpueundrofirgoenFmdexafnmortifipl,dilhing tity be K4hagriraloberphalUanminlid tnkheehypergraph with all four 3-element subsets of a 4-element set yield the same cuts; this was explicitly noted in [12] as a bottleneck to obtain cut-sparsifiers in hypergraphs using CUT queries alone. This inability to learn edges is a hindrance in obtaining low query algorithms in hypergraphs.  \nConnectivity Structure. While the naive binary-search style O (nlog n)-query algorithm (see, e.g, Theorem 5.1 in [16]) to determine connected components works even for hypergraphs, removing the “log n”term is a challenging problem due to the inability of recovering edges due to aforementioned identifiability barrier. Indeed, for graphs random sampling of edges is a key step in the recent O (n)-query randomized algorithms of [3, 29]. Whether one can get an O(n)-query algorithm was was explicitly posed by Chakrabart","cbCaimlZdrxZ5V6R","https://ap.wps.com/l/cbCaimlZdrxZ5V6R","pdf",458916,2,1,20,"English","en",105,"# Abstract\n# Introduction\n## Connectivity Structure\n# Result 1\n# Exact Learnability and Edge Parity\n## Result 2","[{\"question\":\"What problem does CUT-query access address for hypergraphs?\",\"answer\":\"It investigates how much structure an unknown hypergraph can be revealed using CUT queries, specifically targeting connectivity-related information even when edge reconstruction is obstructed.\"},{\"question\":\"How does the paper find connected components of a weighted hypergraph using O(n) expected queries?\",\"answer\":\"It introduces independent families to avoid direct edge learning, then iteratively merges these families using reductions to spanning forest learning in an auxiliary weighted bipartite graph, leveraging known O(n)-query graph connectivity algorithms.\"},{\"question\":\"Why does exact edge learning fail in general for hypergraphs?\",\"answer\":\"Distinct hypergraphs can share identical cut-profiles, creating an identifiability barrier where the CUT function cannot distinguish certain edge sets, making exact edge learning impossible in 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problem does CUT-query access address for hypergraphs?","Question",{"text":75,"@type":76},"It investigates how much structure an unknown hypergraph can be revealed using CUT queries, specifically targeting connectivity-related information even when edge reconstruction is obstructed.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper find connected components of a weighted hypergraph using O(n) expected queries?",{"text":80,"@type":76},"It introduces independent families to avoid direct edge learning, then iteratively merges these families using reductions to spanning forest learning in an auxiliary weighted bipartite graph, leveraging known O(n)-query graph connectivity algorithms.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does exact edge learning fail in general for hypergraphs?",{"text":84,"@type":76},"Distinct hypergraphs can share identical cut-profiles, creating an identifiability barrier where the CUT function cannot distinguish certain edge sets, 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