[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84341-en":3,"doc-seo-84341-105":30,"detail-sidebar-cat-0-en-105":92},{"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},84341,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Optimal Sparsifiers for Abelian Cayley Graphs","For every Cayley graph on a finite abelian group G, an ε-approximate spectral sparsifier exists that preserves Laplacian structure using only O(log |G|) weighted generators, and this dependence is tight. The result yields, for G = F_n^2, an O(n/ε^2)-sized sparsification for F_2-linear codes, improving previous bounds that incurred additional polylog(n) factors. The construction reduces abelian Cayley sparsification to a convex-body volume lower bound, following ideas inspired by ℓ1-sparsifiers.","Optimal Sparsifiers for Abelian Cayley Graphs  \nArpon Basu  \n[arpon.basu@princeton.edu](arpon.basu@princeton.edu)[ ](arpon.basu@princeton.edu)Princeton University  \nPravesh K. Kothari  \n[kothari@cs.princeton.edu](kothari@cs.princeton.edu)[ ](kothari@cs.princeton.edu)Princeton University  \nRaghu Meka‗ [raghum@cs.ucla.edu](raghum@cs.ucla.edu)[ ](raghum@cs.ucla.edu)University of California, Los Angeles  \nStefan Tudose  \n[studose@princeton.edu](studose@princeton.edu)[ ](studose@princeton.edu)Princeton University  \narXiv :2607 .0826 1v 1 [ cs .DS] 9 Jul 2026  \nJuly 10, 2026  \nAbstract  \nWe prove that for every Cayley graph G over any finite abelian group G, there is a weighted Cayley graph with O(log |G|) generators that is a spectral sparsifier for G. This bound is optimal. Applying our bound to the group G = Fn2, yields, as a corollary, O(n/ε2 )-sized code sparsifiers for F2-linear codes, improving on the work of Khanna, Putterman and Sudan [KPS24] who obtained a similar result with an additional polylog(n) loss.  \nOur proof is strongly inspired by a recent work of Reis and Rothvoss [RR26] for the construction of ℓ1-sparsifiers. Following their work, the abelian Cayley sparsification problem can be reduced to establishing a lower bound for the volume of a certain natural convex body. This volume bound follows from a short, elementary argument that relies on character symmetry.  \n1 Introduction  \nSparsification refers to the process of compressing an object (say a graph, or a code, or a set system) while still retaining some essential features of the object. Sparsification was first introduced by Benczúr and Karger [BK96] in the context of cut sparsification, who showed that witmuhltSiilcat(ntih21dεtgefaspwctareosirightsficati,onon, eacouldnd gepreservneralizaetithonesvsaluuceshaosf aslplen cuts of atral sparsigfircaaphtionup[StoT1a1, SS11, BSS14] have proved to be very useful in graph algorithms [BK96, SS11], in solving Laplacian linear systems [ST04], in reducing the space usage of sublinear time algorithms [AGM12, McG14, ADK+ 16, KLM+ 17], and many other applications.  \nGiven the success of sparsification as a paradigm, much effort has also been invested into generalizing graph sparsification to more general objects, such as hypergraphs [KK15, CKN20,  \n‗Supported by NSF EnCORE: Institute for Emerging CORE Methods in Data Science Award \\#2217033 and NSF AF: Small Award \\#2425350  \nKKTY21a, KKTY21b, JLS23, Lee23], codes [KPS24, KPS25, BG25], and CSPs [KK15, FK17, BŽ20], to mention a few applications.  \nOne such generalization which has been investigated is the notion of Cayley sparsification [KPS24, KPS25, HLM+26, BKLM26], which is what we study in this paper.  \nWe now formally define Cayley graphs and Cayley sparsification:  \nDefinition 1.1 (Cayley Graphs). Let G be a group, and let S ⊂ G be a symmetric subset of G, i.e. s ∈ S iff s −1 ∈ S. The Cayley graph Cay(G, S) is a graph on G where g, g′ ∈ G are connected if g −1g′ ∈ S. In general we also consider weighted Cayley graphs, wherein we have a symmetric weight function w : S → R⩾0 (satisfying w(s) = w(s−1) for all s ∈ S), and the weighted Cayley graph Cay(G, S, w) is the graph Cay(G, S) where the edge {g, g′ } ∈ E(Cay(G, S)) receives the weight w(g−1g′) . Thus unweighted Cayley graphs can be viewed as possessing the weight function w : S → {1} .  \nWe can now define Cayley sparsification:  \nDefinition 1.2 (Cayley Sparsification). Let ε ∈ (0 , 1) be some parameter. Given a (weighted) Cayley graph G := Cay(G, S, w), we say Cay(G, S′, w′) is an ε-Cayley sparsifier for G if  \n(1 − ε)L ≼ L′ ≼ (1 + ε)L,  \nwhere L (resp. L′) refers to the Laplacian of Cay(G, S, w) (resp. Cay(G, S′, w′)), and ≼ refers to the Loewner order on the space of Hermitian matrices, i.e. A ≼ B iff B − A is positive semidefinite (PSD) .  \nIf Cay(G, S′, w′) is an ε-Cayley sparsifier for Cay(G, S, w), we write Cay(G, S′, w′) ≈ε Cay(G, S, w) .  \nInvoking spectral graph sparsification primitives such as [SS11, BSS14","cbCaibsxGvkpbSiM","https://ap.wps.com/l/cbCaibsxGvkpbSiM","pdf",422284,5,1,12,"English","en",105,"# Introduction\n## Cayley graphs and Cayley sparsification\n## Comparison to prior Cayley sparsification results\n## Optimal Abelian Cayley Sparsification","[{\"question\":\"What does the paper prove about ε-Cayley sparsifiers for finite abelian groups?\",\"answer\":\"For any Cayley graph over a finite abelian group G, there exists an ε-Cayley sparsifier with only O(log |G|) weighted generators, and the bound is optimal.\"},{\"question\":\"How does the main theorem translate to code sparsification for F2-linear codes?\",\"answer\":\"Applying the result to G = F_n^2 gives an O(n/ε^2)-sized code sparsifier for F_2-linear codes, removing extra logarithmic factors present in earlier work.\"},{\"question\":\"Why do the authors’ techniques relate to convex-body volume lower bounds?\",\"answer\":\"The abelian Cayley sparsification problem is reduced to proving a lower bound on the volume of a natural convex body, supported by an argument using character symmetry.\"}]",1784194944,30,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"optimal-sparsifiers-for-abelian-cayley-graphs","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/optimal-sparsifiers-for-abelian-cayley-graphs/84341/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does the paper prove about ε-Cayley sparsifiers for finite abelian groups?","Question",{"text":76,"@type":77},"For any Cayley graph over a finite abelian group G, there exists an ε-Cayley sparsifier with only O(log |G|) weighted generators, and the bound is optimal.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the main theorem translate to code sparsification for F2-linear codes?",{"text":81,"@type":77},"Applying the result to G = F_n^2 gives an O(n/ε^2)-sized code sparsifier for F_2-linear codes, removing extra logarithmic factors present in earlier work.",{"name":83,"@type":74,"acceptedAnswer":84},"Why do the authors’ techniques relate to convex-body volume lower bounds?",{"text":85,"@type":77},"The abelian Cayley sparsification problem is reduced to proving a lower bound on the volume of a natural convex body, supported by an argument using character 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