[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83504-en":3,"doc-seo-83504-105":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},83504,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Online Matching with Size-Based and Convex Delays","Online min-cost perfect matching with delay (MPMD) addresses m requests arriving over time in a metric space of n points, where an algorithm decides to match or defer each request. The total objective minimizes connection costs (distances of matched request locations) plus delay costs driven by the set of unmatched requests. The work studies two delay models—size-based (MPMD-Size) and convex delays (MPMD-Convex)—and derives improved bounds on competitive ratios, including succinct reductions to metrical task systems for MPMD-Size and constant-competitive algorithms for uniform metrics under broad convex polynomial functions.","arXiv :2607 .00536v 1 [ cs .DS] 1 Jul 2026  \nOnline Matching with Size-Based and Convex Delays  \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 study the online min-cost perfect matching with delay (MPMD) problem where m requests arrive in a metric space of n points. In MPMD, an algorithm can choose to match a request or to delay, and the objective is to minimise the sum of connection and delay costs. The connection cost of a match is the distance between the locations of two matched requests in the metric, and the increase of the delay cost is a function of the set of unmatched requests at every moment. In this paper, we study two different types of delay functions, size-based (MPMD-Size) and convex delays (MPMD-Convex) .  \nThe study of MPMD-Size was initiated by Deryckere and Umboh (APPROX/RANDOM 2023) where the instantaneous delay increment is a non-negative monotone function of the number of unmatched requests. We give an exponential improvement on the lower bounds for the deterministic and randomized competitive ratios. We also give improved upper bounds in terms of n, as opposed to Deryckere and Umboh’s upper bounds that were functions of m, which can be much larger than n. Our results settle the deterministic competitive ratio (up to constants) . At the heart of these results is a succinct encoding scheme of MPMD-Size on a given n-point metric as a metrical task system problem on a 2n−1-point metric.  \nWe also consider MPMD-Convex proposed by Liu et al. (ISAAC 2018) where the delay cost incurred by each request is a uniform convex delay function of the time difference between its arrival time and the moment that it is matched by the algorithm. They focused on delay functions f that are unbounded, non-decreasing, continuous, and satisfy f(0) = f ′(0) = 0, and showed that the deterministic competitive ratio is Ω(n) for n-point uniform metrics. We show that, surprisingly, when f is a non-negative, monotone polynomial with f ′(0) > 0, there is an O(1)-competitive deterministic algorithm for uniform metrics. Our result completes our understanding of MPMD-Convex on uniform metrics for a broad class of functions.  \n2012 ACM Subject Classification Theory of computation → Online algorithms  \nKeywords and phrases Online Algorithms with Delays, Competitive Analysis, Matching  \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 .  \n 1  Introduction  \nIn recent years, there has been much interest in revisiting classical online problems through the lens of online problems with delay (e.g. [3, 17 , 13]) . In such problems, instead of serving  \n2 Online Matching with Size-Based and Convex Delays  \nevery request immediately, one can choose to delay the service of a request while accumulating a delay cost, and the total cost of a solution is the cost of serving all requests plus the accumulated delay costs. In such problems, the trade-off between service cost and delay cost is at the center of the decision making of an online algorithm: the algorithm can wait until many requests have arrived and then apply an offline algorithm to serve them cheaply as a batch, but this will incur a large delay cost.  \nIn this paper, we study the Min-Cost Perfect Matching with Delay (MPMD) problem prop","cbCaihv67Zd9BWcd","https://ap.wps.com/l/cbCaihv67Zd9BWcd","pdf",809780,1,30,"English","en",105,"# Abstract\n# Introduction\n## Problem setting and trade-off\n## Delay cost models","[{\"question\":\"What is the Min-Cost Perfect Matching with Delay (MPMD) problem in the document?\",\"answer\":\"MPMD involves m time-ordered requests in a metric space of n points. The algorithm can match available requests (paying connection cost as a distance) or leave them unmatched (paying delay cost related to pending requests).\"},{\"question\":\"How do connection costs and delay costs differ in MPMD?\",\"answer\":\"Connection cost equals the distance between locations of two matched requests. Delay cost increases according to a delay function determined by the collection of unmatched requests at each moment or by timing differences when convex delays are used.\"},{\"question\":\"What are the two delay models analyzed—MPMD-Size and MPMD-Convex?\",\"answer\":\"MPMD-Size uses instantaneous delay increments that depend monotonically on the number of unmatched requests. MPMD-Convex assigns each request a uniform convex delay based on the time between its arrival and when it is matched.\"}]",1784188483,76,{"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},"online-matching-with-size-based-and-convex-delays","",{"@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/online-matching-with-size-based-and-convex-delays/83504/",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 the Min-Cost Perfect Matching with Delay (MPMD) problem in the document?","Question",{"text":75,"@type":76},"MPMD involves m time-ordered requests in a metric space of n points. The algorithm can match available requests (paying connection cost as a distance) or leave them unmatched (paying delay cost related to pending requests).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do connection costs and delay costs differ in MPMD?",{"text":80,"@type":76},"Connection cost equals the distance between locations of two matched requests. Delay cost increases according to a delay function determined by the collection of unmatched requests at each moment or by timing differences when convex delays are used.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the two delay models analyzed—MPMD-Size and MPMD-Convex?",{"text":84,"@type":76},"MPMD-Size uses instantaneous delay increments that depend monotonically on the number of unmatched requests. MPMD-Convex assigns each request a uniform convex delay based on the time between its arrival and when it is matched.","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},[93,97,101,105,110,115,120,122,127,130,134],{"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":21,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]