[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83250-en":3,"doc-seo-83250-105":30,"detail-sidebar-cat-0-en-105":87},{"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},83250,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries","Studies edge counting in an undirected, unweighted graph under sublinear query access by enriching Independent Set (IS) queries with a random certificate. The proposed oracle, Independent Set with Certificate (ISC), takes a vertex subset and returns a uniformly random edge from the induced subgraph when it is nonempty, otherwise null. A randomized algorithm produces a (1±ε)-approximation to the number of edges with constant success probability using O((log m)^2) queries, yielding exponential separation versus standard IS and global random edge-sampling models. The complexity is also output-sensitive and connects to group testing, decision-versus-counting, and conditional sampling.","arXiv :2607 .07483v 1 [ cs .DS] 8 Jul 2026  \nFrom Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries  \nDebarshi Chanda 1 , Buddha Dev Das 1 , Arijit Ghosh 1 , and Gopinath Mishra2  \n1 Indian Statistical Institute, Kolkata, India  \n2 Institute of Mathematical Sciences, HBNI, Chennai, India  \nAbstract  \nWe study the problem of estimating the number of edges in an undirected, unweighted graph using sublinear query access. We consider a query model that preserves the structure of Independent Set (IS) queries, but augments their output with a random certificate: given a vertex subset, the oracle returns a uniformly random edge from the induced subgraph if one exists, and returns null otherwise.  \nUsing this access, we give a randomized algorithm that outputs a (1±ε)-approximation to the number of edges with constant success probability using (log2 m) queries. This implies an exponential separation from both standard IS queries and global random edge-sampling models: estimating the number of edges using standard IS queries require Ämin ¶ √m,  ©ä  \n‹  \nqueries, while direct random edge-sample access requires Θ( √m) samples. Beyond separation in query complexity, our algorithm is output-sensitive: its query complexity is polylogarithmic in the number of edges in the graph. This aligns with the classical objective in group testing, where one seeks algorithms that are both worst-case optimal and instance-adaptive.  \nConceptually, our model connects group testing, the decision–versus–counting dichotomy, graph property testing, and the “power of a random certificate”, and can be viewed as a structured form of conditional sampling of edges in graphs.  \n1 Introduction  \nWe consider the problem of estimating the number of edges in an undirected, unweighted, simple graph using a sublinear number of queries. Let G = (V, E) be a graph with V =[n] := {1, 2 ,..., n} vertices and |E| = m edges. Given query access to G and an approximation parameter ε ∈ (0 , 1), the objective is to output an estimate  such that  \n(1 − ε)m ≤  ≤ (1 + ε)m  \nwith success probability at least 2/3.  \nQuery model and our result. We work in a query model built on Independent Set (IS) queries, introduced by Beame et al. [6] . The IS query takes a subset of vertices U ⊆ V and returns whether U is an independent set.  \nWe consider the Independent Set with Certificate query, denoted by ISC, which takes as input a subset of vertices U ⊆ V and returns  \n􀀸  \nISC (U) = 􀀼􀀺  \ne ∼ Unif(E(U)), if E(U)  ∅ , 0 , otherwise,  \nwhere E (U) denotes the set of edges in the subgraph of G induced by U, and Unif(S) denotes the uniform distribution over a (nonempty) set S.  \nImportantly, the ISC query preserves the structural constraint of a standard IS query: the algorithm specifies exactly the same vertex subsets as in the IS model, and the only difference lies in the response, which returns a uniformly random edge as a certificate whenever the induced subgraph is nonempty.  \nBeame et al. [6] showed that the number of edges in a graph can be estimated using  \nÅmin ß n2m , √m ™ã = (n2/3)  \nIS queries, establishing a polynomial separation from local query based algorithms that require Θ(n) queries in the worst case [25] . In a subsequent work, Chen et al. [20] nearly resolved the query complexity of edge estimation using IS queries by showing that  \nÅmin ß ~~ ~~ , √m ™ã  \n‹queries are both necessary and sufficient. In particular, this implies a worst-case complexity of Θ( √n) IS queries.  \nOur main result shows that the natural enrichment of IS queries with random certificates yields an exponential separation from standard IS queries for edge estimation, achieving polylogarithmic query complexity in m.  \nTheorem 1 (Main result) . There exists a randomized algorithm that, given access to a graph G through ISC queries and an approximation parameter ε ∈ (0 , 1/3), outputs an estimate  of the number of edges m in G such that, with probability at ","cbCaibE5VVttcJ9A","https://ap.wps.com/l/cbCaibE5VVttcJ9A","pdf",408199,4,1,25,"English","en",105,"# Introduction\n## Problem setting and query access\n## Independent Set with Certificate (ISC) model\n## Main result and query complexity\n## Conceptual connections and group testing perspective","[{\"question\":\"What problem does the document address?\",\"answer\":\"It addresses estimating the number of edges in an undirected, unweighted graph using a sublinear number of queries.\"},{\"question\":\"Why is the result considered an exponential separation?\",\"answer\":\"Because augmenting IS access with random certificates enables polylogarithmic query complexity in m, whereas standard IS queries and global random edge-sampling require significantly more queries (with lower bounds scaling as about √m in the compared models).\"}]",1784186253,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":82,"head_meta":84,"extra_data":86,"updated_unix":28},"from-decision-to-random-certificates-exponential-separation-for-edge-estimation-with-independent-set-queries","",{"@graph":36,"@context":81},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/from-decision-to-random-certificates-exponential-separation-for-edge-estimation-with-independent-set-queries/83250/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the document address?","Question",{"text":75,"@type":76},"It addresses estimating the number of edges in an undirected, unweighted graph using a sublinear number of queries.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is the result considered an exponential separation?",{"text":80,"@type":76},"Because augmenting IS access with random certificates enables polylogarithmic query complexity in m, whereas standard IS queries and global random edge-sampling require significantly more queries (with lower bounds scaling as about √m in the compared models).","https://schema.org",{"og:url":52,"og:type":83,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":85,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":88},[89,93,97,101,106,111,116,119,124,127,131],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Exam",70,"exam",{"id":102,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},5,"Comic",60,"comic",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},6,"Technology",50,"technology",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":117,"slug":118},30,"research-report",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},9,"Religion & Spirituality",20,"religion-spirituality",{"id":122,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":122,"slug":126},"World Cup","world-cup",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":128,"slug":130},10,"Lifestyle","lifestyle",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":102,"slug":134},19,"General","general"]