[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85565-en":3,"doc-seo-85565-105":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},85565,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Min-Sum Set Cover on Parallel Machines","Min-Sum Set Cover models selecting subsets to cover every element while minimizing the total covering times, where an element’s covering time equals the combined costs of sets processed before the first set covering it. The paper extends this classical single-machine scheduling view to m parallel machines, forming the Parallel Min-Sum Set Cover (PMSSC) problem. Approximation results are derived via a Parallel Densest Subfamily subproblem, converted using Maximum Coverage Multiple Knapsack.","arXiv :2604 . 1 1388v4 [ cs .DS] 12 Jul 2026  \nMin-Sum Set Cover on Parallel Machines  \nMichał Szyfelbein  \nAbstract  \nConsider the classical Min-Sum Set Cover problem: We are given a universe U of n elements and a collection S of k subsets of U. Moreover, a cost function is associated with each set. The goal is to find a subsequence of sets from S which covers all elements in U , such that the sum of the covering times of the elements is minimized. The covering time of an element u is the cost of all sets that appear in the sequence before u is first covered. This problem can be seen as a scheduling problem on a single machine, where each job represents a set and elements are represented by some kind of utility that is required to be provided by at least one of the jobs. The goal is to schedule the jobs in such a way to minimize the sum of provision times of the utilities. In this paper we consider a natural generalization of this problem to the case of m machines, processing the jobs in parallel. We call this problem Parallel Min-Sum Set Cover.  \nTo obtain approximation algorithms for various variants of this task, we exploit a crucial subproblem called Parallel Densest Subfamily, where the goal is to find an asignment of sets to the machines that maximizes the ratio of the number of covered elements to the length of the assignment. We prove that an α-approximation algorithm for this problem implies a 4α-approximation algorithm for Parallel Min-Sum Set Cover. Then, we show how to find such an assignment using the well known Maximum Coverage Multiple Knapsack problem. In particular, this yields a e1 + ϵ-approximation for identical machines and an e1 + ϵ-approximation for unrelated machines. If the sets are subject to precedence constraints we give a greedy algorithm for unit cost sets, with an O (k2/3) approximation ratio and an O(log k)-approximation algorithm for out-forest precedence constraints and identical machines. The latter algorithm uses a reduction to the Group Steiner Orienteering problem which is of independent interest.  \nKeywords: Parallel Min Sum Set Cover, Approximation Algorithms, Scheduling  \n1 Introduction  \nConsider the following scenario: A massive infrastructure such as a network of bus stops is required to be built over the course of an extended period of time. The goal of the project is to provide all citizens with access to the public transportation. To do so, it is required to build the stops in such away to enable each citizen to reach at least one of them within a reasonable time. This means, that each such possible stop location covers a certain subset of citizens. The contractor appointed to carryout this task has multiple teams of workers at their disposal, each of which can work on constructing a single stop at any given time. Moreover, the time required to build a stop at a certain location may differ both between places and teams. The goal of the contractor is to schedule the work of the teams in such a way that the average time required for a citizen to get access to the public transportation is minimized. This problem can be modelled using the following generalization of the classical Min-Sum Set Cover problem: We are given a universe U of n elements representing people and a collection S of k subsets of U encoding possible stop locations. There are m machines at our disposal, capable of processing the sets, representing worker teams. Moreover, each set-machine pair is associated with a cost function representing the time required to process the set on that machine. The goal is to find an m-machine schedule of sets from S that covers all elements in U , such that the sum of the covering times of the elements is minimized. The covering time of an element u is the first moment such that some set containing u has already been fully processed. This problem can be seen as a scheduling problem, where each job represents a set and each element is represented by some kind of utility that is requir","cbCaimfI73i0IVxZ","https://ap.wps.com/l/cbCaimfI73i0IVxZ","pdf",532694,4,1,13,"English","en",105,"# Introduction\n## Our results and techniques","[{\"question\":\"What objective does the Min-Sum Set Cover problem optimize?\",\"answer\":\"It selects a subsequence of subsets whose union covers all elements while minimizing the sum of covering times across elements.\"},{\"question\":\"How is Parallel Min-Sum Set Cover (PMSSC) different from the classical version?\",\"answer\":\"PMSSC processes sets on m machines in parallel, and the covering time of an element is defined by when a covering set finishes on its machine.\"},{\"question\":\"What subproblem is central to the paper’s approximation approach?\",\"answer\":\"The paper uses the Parallel Densest Subfamily problem and shows that an α-approximation for it yields a 4α-approximation for Parallel Min-Sum Set Cover.\"}]",1784204632,33,{"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},"min-sum-set-cover-on-parallel-machines","",{"@graph":36,"@context":85},[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/min-sum-set-cover-on-parallel-machines/85565/",{"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",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What objective does the Min-Sum Set Cover problem optimize?","Question",{"text":75,"@type":76},"It selects a subsequence of subsets whose union covers all elements while minimizing the sum of covering times across elements.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is Parallel Min-Sum Set Cover (PMSSC) different from the classical version?",{"text":80,"@type":76},"PMSSC processes sets on m machines in parallel, and the covering time of an element is defined by when a covering set finishes on its machine.",{"name":82,"@type":73,"acceptedAnswer":83},"What subproblem is central to the paper’s approximation approach?",{"text":84,"@type":76},"The paper uses the Parallel Densest Subfamily problem and shows that an α-approximation for it yields a 4α-approximation for Parallel Min-Sum Set Cover.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]