[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85966-en":3,"doc-seo-85966-105":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},85966,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","The Power of Arrival Times in Random-Order Online Facility Location","Online metric facility location with uniform opening costs is analyzed in the random-order model introduced by Meyerson. Previous best results achieved a 3-competitive randomized bound, leaving a gap to the known lower bound of 2. Two improved algorithms are presented: a deterministic method with competitive ratio below 2.42 and a randomized method with competitive ratio below 2.59. The core enhancement uses request arrival time, which encodes geometric density information around each request and improves opening decisions. The randomized approach also preserves the asymptotically optimal O(log n/log log n) adversarial-order guarantee.","The Power of Arrival Times in Random-Order Online Facility Location  \nYichen Huang∗ Harvard University  \nShaofeng H.-C. Jiang† Peking University  \narXiv :2607 . 10564v 1 [ cs .DS] 12 Jul 2026  \nJuly 14, 2026  \nAbstract  \nWe study online metric facility location with uniform opening costs in the random-order model (Meyerson FOCS’01) . The best previous upper bound was a 3-competitive randomized algorithm (Kaplan, Naori, Raz SODA’23), leaving a gap to the best known lower bound of 2. In this work, we give two algorithms with improved competitive ratios: (i) a deterministic algorithm with a competitive ratio below 2 .42 and (ii) a randomized algorithm with a competitive ratio below 2 .59 and the additional property that it retains the asymptotically optimal O(log n/log log n) competitive ratio in the adversarial-order model. A key improvement is to take the arrival time of the request into consideration when making opening decisions: The arrival time carries geometric information about the local density around the request, which fundamentally helps the algorithm.  \n∗ Supported by NSF grant CNS-2107078. Email: [yichenhuang@g.harvard.edu](yichenhuang@g.harvard.edu).†Email: [shaofeng.jiang@pku.edu.cn](shaofeng.jiang@pku.edu.cn).  \n1 Introduction  \nOnline facility location, introduced by Meyerson [Mey01], is a fundamental online optimization problem and has received significant attention in algorithmic study (e.g., [ABUV04; AFGS22; CCMS18; FGG+25; Fot07; Fot11b; GKLX20; KNR23; Mey01]) . In this problem, the algorithm is given in advance a metric space (X, d) and a facility cost parameter f > 0, and it must serve a sequence of request points (v1 ,..., vn) ⊆ X . Upon receiving a request vi, the algorithm may open new facilities at any point in the metric space, paying f opening cost for each opening, and must irrevocably assign the request to an open facility, denoted as yi ∈ X, paying the corresponding metric distance d(vi, yi) as the connection cost. The objective is to minimize the total opening and connection cost. Besides the standard adversarial-order setting [Fot03; Fot07; Mey01], the problem has also been studied in the random-order model [KNR23; Mey01], where the requests are chosen adversarially but the order in which the requests arrive is a uniformly random permutation.  \nWe focus primarily on the random-order model. The best known competitive ratio in the random-order model is 3, achieved by an algorithm belonging to a so-called q-DistProb family [KNR23], which includes Meyerson’s original algorithm [Mey01] whose ratio is 4 . In addition, this algorithm has a unique “good-in-both-worlds” feature: the algorithm can simultaneously achieve an optimal O(log n/log log n) ratio in the adversarial-order setting. Unfortunately, it is unclear if this ratio 3 is tight in general for the random-order setting, since the current lower bound is only 2 [KNR23], which leaves a gap. On the other hand, breaking 3 requires overcoming a fundamental technical barrier, since it has been shown that 3 is the best ratio achievable for any algorithm in the q-DistProb family [KNR23] .  \nIn this work, we break this 3-competitive barrier by devising new families of algorithms beyond q-DistProb. In q-DistProb, for a parameter q > 0, the algorithm maintains the set F of currently open facilities, and when a request x arrives, the algorithm opens a facility at x with probability min{q · d(x, F)/f,1}, after which x is connected to the nearest facility in F.  \nBreaking the Barrier via Arrival Time. The DistProb family makes its decisions based solely on the current distance. We additionally make use of the arrival time of each request. Specifically, we consider a more general family of algorithms, called TimeDist, in which the decision of whether to open a facility at request vt is determined by a function g : (t, d(vt, F)) →7 [0 , 1] describing the probability of opening the request, i.e. ,  \nopen facility at vt with probability g(t, d(vt, F)) . (1)  \nWe ","cbCailcI7R4NQsR9","https://ap.wps.com/l/cbCailcI7R4NQsR9","pdf",535222,3,1,30,"English","en",105,"# Introduction\n## Random-order setting and prior bounds\n## Breaking the 3-competitive barrier via arrival time\n## TimeDist family and Theorem 1.1\n## qt-DistProb family and Theorem 1.2","[{\"question\":\"What problem and model are studied in the document?\",\"answer\":\"The document studies online metric facility location with uniform opening costs in the random-order model, where requests are an adversarial permutation but arrive in uniformly random order.\"},{\"question\":\"What are the main algorithmic improvements reported?\",\"answer\":\"It introduces two algorithms: a deterministic algorithm with competitive ratio below 2.42 and a randomized algorithm with competitive ratio below 2.59.\"},{\"question\":\"How does arrival time help the algorithm decide when to open facilities?\",\"answer\":\"Arrival time is used in a generalized opening probability rule, where the probability depends on both the request’s arrival index and its distance to the current open facilities. This captures geometric information about local density near the request and improves opening decisions.\"}]",1784207441,76,{"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},"the-power-of-arrival-times-in-random-order-online-facility-location","",{"@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/the-power-of-arrival-times-in-random-order-online-facility-location/85966/",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","2026-07-26","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 problem and model are studied in the document?","Question",{"text":75,"@type":76},"The document studies online metric facility location with uniform opening costs in the random-order model, where requests are an adversarial permutation but arrive in uniformly random order.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are the main algorithmic improvements reported?",{"text":80,"@type":76},"It introduces two algorithms: a deterministic algorithm with competitive ratio below 2.42 and a randomized algorithm with competitive ratio below 2.59.",{"name":82,"@type":73,"acceptedAnswer":83},"How does arrival time help the algorithm decide when to open facilities?",{"text":84,"@type":76},"Arrival time is used in a generalized opening probability rule, where the probability depends on both the request’s arrival index and its distance to the current open facilities. This captures geometric information about local density near the request and improves opening decisions.","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},[93,97,101,105,110,115,120,122,127,130,134],{"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":22,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]