[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83542-en":3,"doc-seo-83542-105":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},83542,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Submodular Maximization over Many Matroids via Ordered Local Search","Given a monotone submodular function, the paper studies maximizing a set’s value subject to membership in the intersection of k matroids. It presents a polynomial-time local search algorithm with a (k^2 + o(k)) approximation guarantee, matching the best known unweighted bound. The method greedily orders elements by marginal value and performs α-local swaps with profitability determined by k. Extensions cover Matroid k-Parity and yield improved results for weighted k-Set Packing using a hybrid weight-bucketing strategy.","arXiv :2607 .00843v 1 [ cs .DS] 1 Jul 2026  \nSubmodular Maximization over Many Matroids via Ordered  \nLocal Search  \nNeta Singer ∗ Theophile Thiery †  \nJuly 2, 2026  \nAbstract  \nGiven a monotone submodular function, we consider the problem of finding a maximumvalued set in the intersection of k matroids. Our main result is a polynomial time local search based algorithm achieving a ~~k~~2 + o(k) approximation guarantee. This asymptotically matches the best-known guarantee of ~~k~~2 + ε in the unweighted setting by Lee, Sviridenko, and Vondrák (2009) . Prior to this work, the state-of-the-art was a 1l(4ln)(k2) + o(k)-approximation algorithm obtained by Feldman and Ward (2026) . Our approach extends to Matroid k-Parity yielding the same approximation guarantee.  \nIn contrast to the weight bucketing approach underlying the recent advances of Singer and Thiery (2025) and Feldman and Ward (2026), our algorithm processes elements greedily in decreasing order of marginal value and searches for sufficiently profitable swaps, whose gain exceeds a parameter α given as a function of k. We further combine this idea with the weight bucketing approach to obtain improved guarantees for weighted k-Set Packing. Our second main result is a ln(43)k + o(k)-approximation algorithm for weighted k-Set Packing, improving on the state of the art 2.0~~k~~0561 + O(1)-approximation by Neuwohner (2023) .  \n∗Ecole Polytechnique Fédérale de Lausanne ([Email: neta.singer@epfl.ch](Email: neta.singer@epfl.ch))  \n†Department of Computer Science, ETH Zurich, Zurich,(Email: [theophile.thiery@inf.ethz.ch](theophile.thiery@inf.ethz.ch))  \nContents  \n1 Introduction 3  \n1.1 Overview of our Approach ................................. 4  \n1.2 Techniques ......................................... 4  \n1.3 Paper Organization ..................................... 6  \n2 Preliminaries 6  \n3 Main Algorithm 7  \n3.1 Defining the α-Local Search Subroutine ......................... 7  \n3.2 Defining Ordered α-Local Search ............................. 8  \n4 Preliminaries for the Analysis of Algorithm 2 8  \n4.1 Partitioning the Edges of O by Value ........................... 9  \n4.2 Properties of Incremental Values .............................. 10  \n5 Building Matroid Exchanges 11  \n6 Approximation Guarantee of Algorithm 2 11  \n6.1 Decomposition of the value of O into cardinality bounds ................ 11  \n6.2 Contribution of Appearances ................................ 13  \n6.3 Contribution of Singles ................................... 15  \n6.4 Putting Guarantees Together to Prove Theorem 1 .................... 16  \n7 An improved approximation for set packing via hybrid α-local search and unweighted local search 18  \n7.1 Main Algorithm ....................................... 18  \n7.2 Analysis of Algorithm 4 ................................... 19  \n7.2.1 Partitioning of the optimal solution ........................ 20  \n7.2.2 Negligible contribution of the edges with removed neighborhoods ....... 20  \n7.2.3 Negligible contribution of singles ......................... 21  \n7.2.4 Analyzing the contribution of doubles ....................... 23  \n7.2.5 Approximation guarantee of T, Dfar, and Sfar-unstable .............. 24  \n7.2.6 Putting the pieces together ............................ 26  \n8 Conclusion and Open Questions 27  \nA Matroid Exchanges 31  \nA.1 Constructing a Conflict Graph ............................... 31  \nB Polynomial Runtime 33  \nC Approximation Guarantee Calculations 34  \n1 Introduction  \nMaximizing submodular and linear functions subject to a k-matroid intersection constraint is a central topic in combinatorial optimization. When k = 1, Greedy is optimal for maximizing linear functions, and celebrated algorithms by Calinescu, Chekuri, Pál, and Vondrák [CCPV11] and Filmus and Ward [FW14, BF24] are optimal for monotone submodular functions. For k = 2, Edmonds’matroid intersection algorithm outputs an optimal solution for linear objective functions [Edm03] . Such alg","cbCaiqZ0MRzBiEKE","https://ap.wps.com/l/cbCaiqZ0MRzBiEKE","pdf",704363,3,1,34,"English","en",105,"# Introduction\n## Overview of our Approach\n## Techniques\n# Preliminaries\n# Main Algorithm\n## Defining the α-Local Search Subroutine\n## Defining Ordered α-Local Search\n# Approximation Guarantee and Analysis\n## Building Matroid Exchanges\n## Approximation Guarantee of Algorithm 2\n# Improved Approximation for Set Packing\n## Hybrid α-local search and unweighted local search\n# Conclusion and Open Questions","[{\"question\":\"What optimization problem does the paper address?\",\"answer\":\"It maximizes a monotone submodular function over sets constrained to lie in the intersection of k matroids.\"},{\"question\":\"What is the main performance guarantee of the proposed algorithm?\",\"answer\":\"The algorithm achieves a polynomial-time approximation with ratio k^2 + o(k), matching the best-known unweighted guarantee.\"},{\"question\":\"How does the algorithm decide which elements to process and which swaps to apply?\",\"answer\":\"It processes elements greedily in decreasing order of marginal value, then searches for α-local swaps whose gain exceeds a threshold parameter determined as a function of k.\"}]",1784188714,86,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"submodular-maximization-over-many-matroids-via-ordered-local-search","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/submodular-maximization-over-many-matroids-via-ordered-local-search/83542/",4,{"url":51,"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","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What optimization problem does the paper address?","Question",{"text":75,"@type":76},"It maximizes a monotone submodular function over sets constrained to lie in the intersection of k matroids.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main performance guarantee of the proposed algorithm?",{"text":80,"@type":76},"The algorithm achieves a polynomial-time approximation with ratio k^2 + o(k), matching the best-known unweighted guarantee.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the algorithm decide which elements to process and which swaps to apply?",{"text":84,"@type":76},"It processes elements greedily in decreasing order of marginal value, then searches for α-local swaps whose gain exceeds a threshold parameter determined as a function of 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