[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83550-en":3,"doc-seo-83550-105":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},83550,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Improved Approximation Algorithms for Parallel Task Scheduling and Multiple Cluster Scheduling","Improved approximation algorithms address Parallel Task Scheduling (PTS) by scheduling jobs with fixed processing times and machine requirements to minimize makespan. The work answers an open question by providing an algorithm achieving an approximation bound of (4/3)OPT plus an additive term, with running time on the order of O(n log n). For Multiple Cluster Scheduling (MCS), which distributes identical machines across multiple clusters, the paper improves the running time of a tight 2-approximation and extends the 9/4 approximation to the general case, supported by an implementation demonstrating practical usability.","Improved Approximation Algorithms for Parallel Task Scheduling and Multiple Cluster Scheduling  \nBennet Edler  \n[stu235896@mail.uni-kiel.de](stu235896@mail.uni-kiel.de)[ ](stu235896@mail.uni-kiel.de)Kiel University  \nKiel, Germany  \nFelix Ohnesorge  \n[foh@informatik.uni-kiel.de](foh@informatik.uni-kiel.de)[ ](foh@informatik.uni-kiel.de)Kiel University  \nKiel, Germany  \nKlaus Jansen  \n[kj@informatik.uni-kiel.de](kj@informatik.uni-kiel.de)[ ](kj@informatik.uni-kiel.de)Kiel University  \nKiel, Germany  \nLis Pirotton  \n[lpi@informatik.uni-kiel.de](lpi@informatik.uni-kiel.de)[ ](lpi@informatik.uni-kiel.de)Kiel University  \nKiel, Germany  \narXiv :2607 .00878v 1 [ cs .DS] 1 Jul 2026  \nAbstract  \nIn the problem of Parallel Task Scheduling (PTS), we are asked to schedule 􀀽 jobs, each with a fixed processing time and machine requirement, such that the completion time of the last job is minimized. Jansen and Rau (2019) presented an algorithm for PTS that achieves an approximation ratio of (3/2)OPT + 􀀿max . They additionally posed the open question whether an approximation ratio of (4/3)OPT + 􀀿max is possible. In this work, we present such an algorithm with a running time of 􀀤(􀀽 log 􀀽) .  \nThe problem of Multiple Cluster Scheduling (MCS) is a natural extension of PTS where we are given 􀀣 clusters each of 􀀼 machines to schedule jobs. Jansen and Rau (2019) adapted their PTS algorithm to MCS with the following results: (1) a 2 approximation, and (2) a near-linear 9/4 approximation if 􀀣 is divisible by 3 . We improve the running time of their 2-approximation and generalize the 9/4 approximation to the general case. The 2-approximation for MCS is tight, since one cannot hope for an approximation ratio better than 2, unless P=NP [Zhuk, 2006] . In addition to our theoretical results, we implement our algorithm and show its practical applicability.  \nCCS Concepts  \n• Theory of computation → Approximation algorithms analysis; Scheduling algorithms.  \nKeywords  \nScheduling, near linear time algorithm  \n1 Introduction  \nIn this paper, we consider the problem of Parallel Task Scheduling (PTS) where one is given a set J = {1, . . . , 􀀽 } of jobs and 􀀼 identical machines. Each job 􀀹 ∈ J is specified by a processing time 􀀿 􀀹 ∈ Q + and a machine requirement 􀁀 􀀹 ∈ N ≤􀀼 . All jobs shall be executed non-preemptively and not necessarily contiguously on 􀀼 identical machines. A solution (schedule) is a function 􀁦 : J ↦→ Q + , assigning starting times to each job. Feasible schedules are required to satisfy the resource constraint, that is, at any time 􀁧 ∈ Q + the total number of machines allocated to concurrently running jobs  \ndoes not exceed 􀀼 :  \n∑︁ 􀁀 􀀹 ≤ 􀀼 .  \n􀀹 ∈ J  \n􀁦 ( 􀀹 ) ≤􀁧\u003C􀁦 ( 􀀹 )+􀀿 􀀹  \nThe objective is to find a schedule with minimal height (makespan), defined as ℎ(􀁦) = max􀀹 ∈ J(􀁦(􀀹) + 􀀿 􀀹) .  \nThe PTS problem arises in a variety of applications, including batch processing, multiprocessor job scheduling and resource allocation in high-performance and cloud computing systems. From a theoretic point of view, the problem is known to be strongly 􀀣 􀀥-hard, even when the number of machines is a small constant (􀀼 ≥ 4) [8, 10] . As exact polynomial-time algorithms are therefore unlikely to exist, the focus of research has shifted toward the development of approximation algorithms. Those approaches aim to compute feasible schedules with bounded makespan relative to the optimum. More precisely, an algorithm is said to be an 􀁕-approximation algorithm with 􀁕 ≥ 1 if for every instance 􀀞 , the height ℎ(􀀖(􀀞 )) of the resulting schedule satisfies ℎ (􀀖(􀀞 )) ≤ 􀁕OPT (􀀞 ), where OPT (􀀞 ) denotes the optimal makespan for instance 􀀞 . The approximation ratio may consist of both a multiplicative 􀁕 ≥ 1 and an additive 􀁖 ≥ 0 term. Then the height 􀀖(􀀞 ) satisfies 􀀖(􀀞 ) ≤ 􀁕OPT (􀀞 ) + 􀁖 for all instances 􀀞 .  \nA natural generalization of PTS is the Multiple Cluster Scheduling (MCS) problem. Here, the computational resources are distributed across 􀀣 ≥ 1 disjoint clusters, each co","cbCaigov8Qe5RJwx","https://ap.wps.com/l/cbCaigov8Qe5RJwx","pdf",932291,5,1,11,"English","en",105,"# Introduction\n## Related work","[{\"question\":\"What problem does this paper study?\",\"answer\":\"The paper studies Parallel Task Scheduling (PTS) and its extension, Multiple Cluster Scheduling (MCS), both aiming to minimize makespan.\"},{\"question\":\"What improvement is achieved for PTS approximation?\",\"answer\":\"It provides an algorithm with approximation ratio (4/3)OPT plus an additive term, improving upon prior results and resolving the posed open question.\"},{\"question\":\"How are results extended to MCS and what about tightness?\",\"answer\":\"For MCS, it improves the running time of the 2-approximation and generalizes the 9/4 approximation beyond special cases; the 2-approximation is tight unless P=NP, and the paper also includes an implementation.\"}]",1784188762,28,{"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},"improved-approximation-algorithms-for-parallel-task-scheduling-and-multiple-cluster-scheduling","",{"@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/improved-approximation-algorithms-for-parallel-task-scheduling-and-multiple-cluster-scheduling/83550/",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","2026-07-26","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 problem does this paper study?","Question",{"text":76,"@type":77},"The paper studies Parallel Task Scheduling (PTS) and its extension, Multiple Cluster Scheduling (MCS), both aiming to minimize makespan.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What improvement is achieved for PTS approximation?",{"text":81,"@type":77},"It provides an algorithm with approximation ratio (4/3)OPT plus an additive term, improving upon prior results and resolving the posed open question.",{"name":83,"@type":74,"acceptedAnswer":84},"How are results extended to MCS and what about tightness?",{"text":85,"@type":77},"For MCS, it improves the running time of the 2-approximation and generalizes the 9/4 approximation beyond special cases; the 2-approximation is tight unless P=NP, and the paper also includes an implementation.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":20,"slug":138},19,"General","general"]