[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83354-en":3,"doc-seo-83354-105":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},83354,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Primal-Dual Online Algorithms for the Parking Permit Problem","The paper revisits the Parking Permit Problem (PPP), a classic online problem generalizing the ski rental setting, using the primal-dual framework. It develops simple deterministic and randomized algorithms that achieve improved performance guarantees without relying on reductions to the laminar variant, which can degrade competitive ratios. The work provides near-matching lower bounds, computes the deterministic competitive ratio exactly, and establishes the randomized competitive ratio up to an additive constant.","arXiv :2607 .08262v 1 [ cs .DS] 9 Jul 2026  \nPrimal-Dual Online Algorithms for the Parking Permit Problem  \nChristian Coester \\#   \nDepartment of Computer Science, University of Oxford, United Kingdom Alex Turoczy \\#   \nDepartment of Computer Science, University of Oxford, United Kingdom  \n~~ Abstract ~~  \nThe Parking Permit Problem (PPP), first studied by Meyerson, is a classic online problem generalizing the ski rental problem. We re-examine the PPP using the primal-dual scheme, obtaining simple algorithms with superior performance guarantees. Unlike previous work, which relied on reductions that degraded competitive ratios, we work with the problem’s structure directly. We also provide near-matching lower bounds. Using the primal-dual framework, we find the PPP’s deterministic competitive ratio exactly, and the randomized competitive ratio within an additive constant.  \n2012 ACM Subject Classification Theory of computation → Online algorithms  \nKeywords and phrases Online Algorithms, Competitive Analysis, Primal-Dual Algorithms, Parking Permit Problem  \nFunding Funded by the European Union (ERC, CCOO, 101165139) . Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can beheld responsible for them.  \n 1  Introduction  \nThe Parking Permit Problem (PPP) is a fundamental problem in online algorithms. It is the seminal leasing problem, meaning an algorithm may purchase a resource to satisfy constraints, but where the purchases last a fixed contiguous time duration, regardless of whether it is used in this time. In this paper, we reinvestigate this classic problem via the primal-dual framework, obtaining new and simpler algorithms with improved performance guarantees.  \nThe PPP can be defined by the following canonical parking permit scenario. Suppose that you walk to work when the weather is good, and drive when it rains. When you drive, you must possess a valid parking permit, but there are K different types. Each permit type lastsa different duration and expires on a fixed day regardless of how many times it is utilized, where longer permits are cheaper on a per-day basis. The dilemma in the PPP is: what permit purchases should you make?  \nEssentially all prior work on PPP and variants [24, 11] involve a reduction to what we shall call the Laminar PPP. In the Laminar PPP, the permit cannot just be purchased starting from any arbitrary day. Instead, the permit durations form a laminar set family, meaning that two different permits’ durations either do not overlap, or one is contained inside the other (see Figure 1) . The reduction to the Laminar PPP loses a constant factor in the competitive ratio. In this paper, we provide techniques for solving leasing problems without assumptions of laminarity, leading to more direct algorithms with superior performance.  \n1.1 Our Results  \nUsing primal-dual algorithms, we prove new upper and lower bounds for both deterministic and randomized algorithms. Our results in comparison to previous work are summarized in  \n2 Primal-Dual Online Algorithms for the Parking Permit Problem  \n~~ ~~ 4  \n~~ ~~ ~~ ~~ ~~ ~~ 3  \n~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ 2  \n~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ 1  \n  1            12   Time  \n Figure 1 Illustration of the Laminar PPP with permit durations of length 1 , 2 , 4 and 12.  \nTable 1 .  \n\n|  |  | Deterministic PPP | Randomized PPP |\n| --- | --- | --- | --- |\n| Upper bounds | Previous New | 4K [24]\u003Cbr>K | 16 ln K [24] ln K + ln ln K + O(1) |\n| Lower bounds | Previous New | K/6 [24], 3 for K = 3 [26]\u003Cbr>K | log2 (K)/2 [24] ln K + ln ln K + o(1) |\n\n Table 1 Summary of prior and new upper and lower bounds for the PPP.  \nIn Section 2, we prove that the deterministic competitive ratio is exactly K, closing a factor 24 gap between prior upper and lower bounds. Our algor","cbCaieXkAYsGBwGO","https://ap.wps.com/l/cbCaieXkAYsGBwGO","pdf",552251,1,16,"English","en",105,"# Introduction\n## Our Results\n## Related Work\n### The Parking Permit Problem","[{\"question\":\"What is the Parking Permit Problem (PPP) and why is it important in online algorithms?\",\"answer\":\"The PPP models a purchasing dilemma where an algorithm must buy permits that remain valid for a fixed contiguous time duration. It generalizes the ski rental problem and serves as a fundamental benchmark for competitive analysis of online algorithms.\"},{\"question\":\"What is the main methodological contribution of the paper?\",\"answer\":\"The paper applies the primal-dual scheme directly to the non-laminar version of the PPP. This avoids competitive-ratio loss caused by reductions to the laminar PPP and addresses unbounded row-sparsity in the associated linear program.\"},{\"question\":\"What competitive ratios does the paper achieve for deterministic and randomized algorithms?\",\"answer\":\"For deterministic algorithms, the competitive ratio is exactly K. For randomized algorithms, the competitive ratio is at most ln K + ln ln K + O(1), and the paper also provides a matching lower bound up to an additive constant.\"}]",1784186960,40,{"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},"primal-dual-online-algorithms-for-the-parking-permit-problem","",{"@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/primal-dual-online-algorithms-for-the-parking-permit-problem/83354/",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 Parking Permit Problem (PPP) and why is it important in online algorithms?","Question",{"text":75,"@type":76},"The PPP models a purchasing dilemma where an algorithm must buy permits that remain valid for a fixed contiguous time duration. It generalizes the ski rental problem and serves as a fundamental benchmark for competitive analysis of online algorithms.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the main methodological contribution of the paper?",{"text":80,"@type":76},"The paper applies the primal-dual scheme directly to the non-laminar version of the PPP. This avoids competitive-ratio loss caused by reductions to the laminar PPP and addresses unbounded row-sparsity in the associated linear program.",{"name":82,"@type":73,"acceptedAnswer":83},"What competitive ratios does the paper achieve for deterministic and randomized algorithms?",{"text":84,"@type":76},"For deterministic algorithms, the competitive ratio is exactly K. For randomized algorithms, the competitive ratio is at most ln K + ln ln K + O(1), and the paper also provides a matching lower bound up to an additive constant.","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},[93,97,101,105,110,115,119,122,127,130,134],{"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":28,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"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"]