[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84316-en":3,"doc-seo-84316-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84316,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Approximation Algorithms for Matroidal Prerequisite Systems","Optimal decision making often depends on prerequisites, but such dependencies may reflect functional capabilities rather than literal items, allowing a required capability to be provided by one or more interacting alternatives. This work defines matroidal prerequisite systems (MPS), combining a poset for ordered prerequisite relations with a matroid that determines when those prerequisites are satisfied through span and independence. The paper develops approximation algorithms for additive and monotone submodular maximization over feasible words, with guarantees governed by structural parameters of the poset–matroid interaction.","arXiv :2607 .08 15 1v 1 [ cs .DS] 9 Jul 2026  \nApproximation Algorithms for Matroidal Prerequisite Systems  \nRobert P. Streit∗ Vijay K. Garg∗ ,†  \nMay 6, 2026  \nAbstract  \nOptimal selections in a decision process are often constrained by prerequisites. However, such prerequisites can encode functional rather than literal dependencies, so a required dependency may be supplied by one or several interacting alternatives. We introduce matroidal prerequisite systems (MPS), a combinatorial constraint structure where a poset specifies prerequisites while a matroid determines when those prerequisites have been satisfied by its span. This createsan order-sensitive notion of feasibility over words, where feasible words are associated with independent sets, while dependencies may be fulfilled through substitutable functionality.  \nOur main contribution is approximation algorithms for nonnegative additive maximization and monotone submodular maximization over the feasible words of an MPS. The guarantees are determined by two structural parameters, namely the maximum matroid rank ∆ of a principal ideal in the poset and the maximum matroid connectivity λmax. These parameters measure the distance an MPS is from encoding a matroid or a poset antimatroid, respectively, both of which are generalized by an MPS. For additive maximization, we obtain efficient deterministic ∆-and (1+λmax)-approximation algorithms. By extending these techniques, we obtain efficient deterministic (2 + λmax)-approximation and randomized (∆2 · (1 − 1/e − δ)−1)-approximation algorithms for all δ > 0 for submodular maximization. The algorithm design and analysis use the theory of polymatroid greedoids, through a cryptomorphism we prove between an MPSand a strong polymatroid greedoid. Finally, an approximation-preserving reduction from densest k-subgraph shows it is not possible to efficiently compute a min{∆,λmax }o(1)-approximation to nonnegative additive maximization over the feasible words of an MPS under the Gap Exponential Time Hypothesis. Thus, an MPS provides a tractable, but provably nontrivial, framework for combinatorial optimization with interacting prerequisites, independence, and substitution.  \n1 Introduction  \nIn a discrete decision process, some decisions are not meaningful, feasible, or useful until other functionality is already present. This is perhaps most obvious in task scheduling problems, where dependencies impose constraints on what can and cannot be completed. However, such dependencies are rarely literal. Some project task may require not an earlier task itself, but rather some capability, interface, or information that earlier tasks are meant to provide. These required functionalities maybe satisfied by alternative selections of one or many interacting decisions. Yet, existing prerequisite models (e.g. [Pic76; JN83; SY00; McC+17; KS07]) typically require prerequisites to be selected explicitly; i.e. if y is a prerequisite of z, then z can only be chosen after y. So, there is no awareness of the overlap or substitutable nature of the functionalities of the decisions.  \n∗ Department of Electrical and Computer Engineering at The University of Texas at Austin.  \n†Partially supported by NSF CNS-1812349, and the Cullen Trust for Higher Education Endowed Professorship.  \nMatroids are set families encoding such a substitution structure. Recall, a matroid M is anonempty set family, whose members are called independent, such that every subset of an independent set is also independent and for every independent X and Y with |Y | > |X| there exists z ∈ Y \\ X such that X + z is independent. The rank rM (X) of an arbitrary set X is the size of a maximal independent set it contains, while the span σM (X) is the union of X with the elements which do not increase its rank. Now, if feasible sequences of decisions x 1 . . . xℓ are always such that {x1 ,..., xℓ} is independent, then the span σM ({x1 ,..., xℓ}) can encode the functionalities covered by x 1 . . . xℓ","cbCailxN6rwiXsDy","https://ap.wps.com/l/cbCailxN6rwiXsDy","pdf",713238,1,33,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What is a matroidal prerequisite system (MPS)?\",\"answer\":\"An MPS combines a poset specifying prerequisite relations with a matroid that determines when prerequisite coverage is satisfied via span and maintained by independence. Feasibility depends on the order-sensitive interaction between prerequisites and matroidal substitution.\"},{\"question\":\"Which optimization problems does the paper focus on?\",\"answer\":\"It develops approximation algorithms for nonnegative additive maximization and for monotone submodular maximization over the feasible words defined by an MPS.\"},{\"question\":\"How are the approximation guarantees determined?\",\"answer\":\"The guarantees depend on structural parameters: the maximum matroid rank ∆ of a principal ideal in the poset and the maximum matroid connectivity λmax. These parameters quantify how far the MPS is from degenerating into a matroid or a poset antimatroid.\"}]",1784194788,83,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"approximation-algorithms-for-matroidal-prerequisite-systems","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/approximation-algorithms-for-matroidal-prerequisite-systems/84316/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 is a matroidal prerequisite system (MPS)?","Question",{"text":75,"@type":76},"An MPS combines a poset specifying prerequisite relations with a matroid that determines when prerequisite coverage is satisfied via span and maintained by independence. Feasibility depends on the order-sensitive interaction between prerequisites and matroidal substitution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which optimization problems does the paper focus on?",{"text":80,"@type":76},"It develops approximation algorithms for nonnegative additive maximization and for monotone submodular maximization over the feasible words defined by an MPS.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the approximation guarantees determined?",{"text":84,"@type":76},"The guarantees depend on structural parameters: the maximum matroid rank ∆ of a principal ideal in the poset and the maximum matroid connectivity λmax. These parameters quantify how far the MPS is from degenerating into a matroid or a poset antimatroid.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]