[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81680-en":3,"doc-seo-81680-105":30,"detail-sidebar-cat-0-en-105":83},{"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},81680,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Sparsify Submodular Functions under Cardinality Constraints","Submodular sparsification generalizes classical sparsification from graphs and matrices to sums of submodular functions. For a sum F(S)=f1(S)+…+fm(S) with fi:{0,1}n→R≥0, the goal is a small weighted sparsifier matching F(S) for all subsets S under a cardinality restriction. The work addresses how limiting |S|≤k can reduce sparsifier size: it gives an O(nk2 log n) construction for arbitrary submodular functions, improving over the general Ω(n2) bound when k=o(√n), and derives strong lower bounds for natural families.","arXiv :2606 .20777v2 [ cs .DS] 10 Jul 2026  \nSparsify Submodular Functions under Cardinality Constraints  \nZhengting Bao∗ Dongrun Cai† Xue Chen‡  \nJuly 13, 2026  \nAbstract  \nSubmodular sparsification is a generalization of classical sparsification problems from graphsand matrices to sums of submodular functions. Given a sum F (S) := f1 (S) + ··· + fm (S) of m submodular functions f1 ,..., fm : {0, 1}n → R ≥0, a size-s sparsifier of F is a weight vector w ∈ R0 with at most s non-zero entries such that w 1 f1 (S) + ··· + wmfm (S) ≈ F (S) for any subset S ⊆ [n] . Motivated by the broad applications of submodular functions in data mining and economics, submodular sparsification has been studied in the past few years. For general submodular functions, Kenneth and Krauthgamer provided an efficient constructio˜n of  \nsize O (n3 ) . Although several families of submodular functions admit sparsifiers of size O (n), the authors of [CKP+ 17] proved an Ω(n2 ) lower bound on the size of sparsifiers for the general case.  \nIn this work, we study how much a cardinality constraint, such as restricting S to subsets of size at most k, can reduce the size of submodular sparsifiers. Specifically, if the guaranty is w 1 f1 (S) + ··· + wmfm (S) ≈ F (S) for any subset S ⊆ [n] of cardinality at most k, are there sparsifiers of size o(n2 )? Our main result shows an efficient sparsifier of size O(nk2 log n) for sums of arbitrary submodular functions. This beats the Ω(n2 ) lower bound for the general setting when k = o ( √n) and leaves a factor of about k to the lower bound Ω(kn) . Next, we study which types of submodular functions admit sparsifiers of size (k log n)O(1) under cardinality constraints. Our main result provides strong lower bounds for several natural families.  \nOur sparsificaiton algorithms combine several techniques to extend the importance-sampling framework, including the Lov´asz extension, Edmonds’ greedy algorithm, the bicriteria approximation of submodular minimization, and Kenneth and Krauthgamer’s approach. In particular, we give an efficient algorithm to obtain a tight estimate (up to a constant) for the sensitivity of each fi under cardinality constraints.  \n∗[bztminamoto@mail.ustc.edu.cn](bztminamoto@mail.ustc.edu.cn) , University of Science and Technology of China, Hefei 230026, China .  \n† [cdr@mail.ustc.edu.cn](cdr@mail.ustc.edu.cn) , University of Science and Technology of China, Hefei 230026, China .  \n‡[xuechen1989@ustc.edu.cn](xuechen1989@ustc.edu.cn) , University of Science and Technology of China, Hefei 230026, China and Hefei National Laboratory, Hefei 230088, China . Supported by NSFC 62372424 and Innovation Program for Quantum Science and Technology 2021ZD0302901 .  \n1 Introduction  \nSparsification is a powerful tool in algorithm design. Its goal is to reduce the size of data efficiently while retaining some key properties. Two notable directions are graph sparsification and matrix sparsification. For graphs, the seminal work of Bencz´ur and Karger [BK96] provided an efficient construction of cut sparsifiers: for an undirected graph G = (V, E), let the cut function of G on a subset S be cutG (S) = Pe∈E cute(S) where cute(S) ∈ {0, 1} indicates whether S cuts the edge e or not; its algorithm outputs a cut sparsifier—a subset E ′ ⊆ E with weights w—of size O (nlog n) such that  \nX we · cute(S) ≈ cutG (S) for every S.  \ne∈E′  \nA stronger notion, introduced by Spielman and Teng [ST04], is a spectral sparsifier that preserves the spectrum of the Laplacian matrix of the graph G. Its generalizations include hypergraphs [KK15, CKN20, KKTY21, Lee23, JLS23], directed hypergraphs [SY19, VBK23, OST23, KPS24a], CSP sparsification [BZ20, KPS25], and code sparsification [KPS24b, BG25, HLM+26] .  \nOn the other hand, matrix sparsification studies how to reduce the number of rows in a matrix A ∈ Rm ×n while preserving a certain norm for vectors in the form of Ax. Two classical examples are ℓp subspace embeddings and the restricted is","cbCaillpksNLDMsn","https://ap.wps.com/l/cbCaillpksNLDMsn","pdf",533347,4,1,37,"English","en",105,"# Abstract\n# Introduction\n## Sparsification background: graphs and matrices\n## Submodular functions and previous results\n## Goal under cardinality constraints","[{\"question\":\"What are the main proof ideas used by the paper’s algorithms?\",\"answer\":\"The algorithms extend an importance-sampling framework by combining tools such as the Lovász extension, Edmonds’ greedy algorithm, bicriteria approximation for submodular minimization, and an approach from Kenneth and Krauthgamer, including efficient estimation of sensitivity under cardinality constraints.\"}]",1784175379,93,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"sparsify-submodular-functions-under-cardinality-constraints","",{"@graph":36,"@context":77},[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/sparsify-submodular-functions-under-cardinality-constraints/81680/",{"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","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What are the main proof ideas used by the paper’s algorithms?","Question",{"text":75,"@type":76},"The algorithms extend an importance-sampling framework by combining tools such as the Lovász extension, Edmonds’ greedy algorithm, bicriteria approximation for submodular minimization, and an approach from Kenneth and Krauthgamer, including efficient estimation of sensitivity under cardinality constraints.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]