[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83190-en":3,"doc-seo-83190-105":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},83190,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Improved Algorithms and Lower Bounds for Parametrized Metrical Service Systems","The document studies the parametrized version of the classical metrical service system (MSS) online problem, where the adversary’s request sequence uses at most m distinct request types known to the algorithm in advance. It develops tight results in terms of m: a deterministic O(m)-competitive algorithm for MSS on weighted stars via an interval covering formulation and primal-dual method, matching an Ω(m) lower bound. It also strengthens lower bounds on hierarchically separated trees (HSTs), proves impossibility results for O(1)-competitive algorithms for m≥4, and characterizes cases m=2 and m≥3 on general metrics.","arXiv :2607 .07098v 1 [ cs .DS] 8 Jul 2026  \nImproved Algorithms and Lower Bounds for Parametrized Metrical Service Systems  \nJunhao Gan \\#   \nSchool of Computing and Information Systems, The University of Melbourne, Australia Xiao Sun \\#   \nSchool of Computing and Information Systems, The University of Melbourne, Australia Seeun William Umboh \\#   \nSchool of Computing and Information Systems, The University of Melbourne, Australia  \nARC Training Centre in Optimisation Technologies, Integrated Methodologies, and Applications (OPTIMA), Australia  \n~~ Abstract ~~  \nWe consider the parametrized setting of the classical metrical service system (MSS) problem first studied by Bubeck and Rabani (APPROX/RANDOM 2020) . In this setting, the adversary is restricted to a set of m distinct request types, known to the algorithm in advance. The goal is to obtain competitive ratio bounds in terms of m. In this work, we make significant progress in understanding the landscape of parametrized MSS and resolve several open problems from Bubeck and Rabani.  \nOur first main result is a tight bound for parametrized MSS on weighted stars. Previously, Bubeck and Rabani gave a randomized lower bound of Ω(m) and deterministic upper bound of O(2m ) . We show that, surprisingly, a deterministic O (m)-competitive algorithm exists, matching the randomized lower bound. Our key insight is an interval covering formulation of MSS on weighted stars which enables an application of the primal-dual method.  \nOur second main contribution is an improved lower bound construction for parametrized MSSon hierarchically separated trees (HSTs) . Bubeck and Rabani’s construction gave a ω(1) lower bound when m ≥ 6. Our improved lower bounds are tight for 2-level HSTs and also rule out O(1)-competitive algorithms on HSTs when the parameter m ≥ 4. We also complement these results by giving a deterministic O(1)-competitive algorithm on general metrics when m = 2 while showing that it is impossible when m ≥ 3.  \n2012 ACM Subject Classification Theory of computation → Online algorithms  \nKeywords and phrases online algorithms, competitive analysis, metrical service systems, metrical task systems  \nFunding Junhao Gan: Supported in part by the Australian Government through the Australian Research Council ARC DP230102908 .  \nXiao Sun: Supported by the Australian Government through the Australian Research Council DP240101353 and by the University of Melbourne through the Melbourne Research Scholarship. Seeun William Umboh: Supported by the Australian Government through the Australian Research Council DP240101353 .  \nAcknowledgements We thank Christian Coester for insight into the case of m = 3 on general metrics.  \n 1  Introduction  \nThe metrical task system (MTS) problem introduced by Borodin et al. [2] is a general framework that models a wide range of online computing problems, e.g., paging [15] and k-server [13] . In this problem, one server moves between different points (also called states) in an n-point metric space to process a sequence of requests that arrive one by one; each  \n2 Improved Algorithms and Lower Bounds for Parametrized Metrical Service Systems  \nrequest incurs different costs when the server is at different states. The total cost is the sum of the transition cost of moving between different states and the cost of serving the requests. An important special case is the metrical service system (MSS) problem [9] where every request has cost either 0 or infinity everywhere in the whole metric space. Note that paging and k-server are both MSS problems.  \nSince its introduction almost four decades ago, MTS has been intensively studied on both general metrics and some special classes of metrics. On general metrics, in a series of recent breakthrough results, Bubeck et al. [4] showed that the randomized competitive ratio is O (log2 n) and Bubeck et al. [3] gave a matching existential lower bound 1 , refuting the previous widely-believed conjecture of the existence of a ","cbCairzrKF4SnYje","https://ap.wps.com/l/cbCairzrKF4SnYje","pdf",631814,3,1,23,"English","en",105,"# Introduction\n## MTS and MSS background\n## Parametrized request-type setting\n## Motivation for parametrization","[{\"question\":\"What is the parametrized MSS model considered in the document?\",\"answer\":\"The adversary is restricted to m distinct request types, and this set is known to the algorithm in advance. Competitive ratio bounds are derived as functions of m.\"},{\"question\":\"What is the main algorithmic result for weighted stars?\",\"answer\":\"The work gives a deterministic O(m)-competitive algorithm for parametrized MSS on weighted stars. It matches a randomized Ω(m) lower bound and uses an interval covering formulation with a primal-dual method.\"},{\"question\":\"How do the paper’s lower bounds change on HSTs and general metrics?\",\"answer\":\"Improved lower bounds are tight for 2-level HSTs and rule out O(1)-competitive algorithms on HSTs when m≥4. It also provides a deterministic O(1)-competitive algorithm on general metrics for m=2, while showing impossibility for m≥3.\"}]",1784185849,58,{"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},"improved-algorithms-and-lower-bounds-for-parametrized-metrical-service-systems","",{"@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/improved-algorithms-and-lower-bounds-for-parametrized-metrical-service-systems/83190/",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","2026-07-25","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 the parametrized MSS model considered in the document?","Question",{"text":75,"@type":76},"The adversary is restricted to m distinct request types, and this set is known to the algorithm in advance. Competitive ratio bounds are derived as functions of m.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main algorithmic result for weighted stars?",{"text":80,"@type":76},"The work gives a deterministic O(m)-competitive algorithm for parametrized MSS on weighted stars. It matches a randomized Ω(m) lower bound and uses an interval covering formulation with a primal-dual method.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the paper’s lower bounds change on HSTs and general metrics?",{"text":84,"@type":76},"Improved lower bounds are tight for 2-level HSTs and rule out O(1)-competitive algorithms on HSTs when m≥4. It also provides a deterministic O(1)-competitive algorithm on general metrics for m=2, while showing impossibility for m≥3.","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":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":52,"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"]