[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83565-en":3,"doc-seo-83565-105":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},83565,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Online Fair Division Meets Reordering Buffers","Online fair division of indivisible mixed manna is studied under additive valuations, where items arrive sequentially and must be allocated irrevocably while agents assign positive, negative, or zero values. The work targets temporal fairness via envy-freeness (EF) and envy-freeness up to one item (EF1). Since standard online settings have strong negative results, reordering buffers are introduced to store and rearrange a limited number of items, bridging online and offline extremes.","arXiv :2607 .0 1 159v 1 [ cs .GT] 1 Jul 2026  \nOnline Fair Division Meets Reordering Buffers  \nGeorgios Amanatidis1,2 , Giulio Giaconi3 , Evangelos Markakis1,2,4 , and Nicos Protopapas1,2  \n1Athens University of Economics and Business, Athens, Greece 2Archimedes/Athena RC, Athens, Greece  \n3HSBC Holdings Plc., United Kingdom  \n4Input Output Global (IOG), Athens, Greece  \nJuly 2, 2026  \nAbstract  \nWe study the online fair division of indivisible mixed manna among agents with additive valuation functions. Under the standard online model, at each time step an indivisible item arrives; each agent may assign it a positive, negative, or zero value, and it must be irrevocably allocated, before the arrival of the next item. At the same time, we also wish to maintain some fairness guarantee, and in this work we focus on envy-freeness (EF) and one ofits most prominent relaxations, envy-freeness up to one item (EF1) . Given the strong negative and the scarce positive results for this problem without additional assumptions, we augment our algorithms with buffers that can store and rearrange a limited number of items. This setting interpolates naturally between the fully online case (no buffer) and the fully offline case (a buffer large enough to hold all items). We show that algorithms equipped with reasonably sized buffers can achieve strong guarantees for personalized k-value instances, i.e., instances in which each agent assigns at most k distinct values to items. In particular, we construct allocations that are EF1 at every time step and EF at most time steps, using a buffer of size linear in k and in the number of agents. Our approach relies on novel combinatorial arguments and on constructing a sequence of envy-free matchings that allocates most items. Finally, we extend our results to general additive valuation functions, with a dependence on the largest per-agent ratio between two values of the same sign, and we also identify limitations of our approach via impossibility results on the use of buffers with smaller size.  \n1 Introduction  \nOur work concerns the fair allocation of indivisible items to a set of interested agents. Fair division has attracted significant interest within the broader algorithmic game theory community, with a sizeable volume of recent literature, as can also be seen by surveys such as Amanatidis et al. [2023], Liu et al.[2024], Suksompong [2021] and Biswas et al. [2023] . The emergence of further motivating applications, including among others food donation programs Mertzanidis et al. [2024], further contributes to the growing momentum of the relevant community. This has naturally led to a variety offair division models, dependent on the type of items to be allocated, the type of preferences, but also on possible constraints on the allocation space and the fairness notions of interest.  \nIn this work, we consider an online scenario where the items are not available from the beginning but instead arrive sequentially, one by one. This can be seen as a more realistic model, compared to the more commonly studied offline model, and is motivated by scheduling applications and other problems where resources are  \nreleased over time. Therefore, an algorithm under this model needs to maintain a partial allocation that is being updated as time progresses, until there are no further arrivals. Furthermore, regarding the type of goods, we focus on the most general setting that is commonly referred to as mixed manna, where an item can be valued either non-negatively (perceived as a good) or non-positively (perceived as a chore) by an agent. Finally, our target fairness notions are envy-freeness (EF) and one of its most prominent relaxations, envy-freeness up to one item (EF1) . Given these considerations, ideally we would like to have algorithms that maintain temporal fairness, i.e., the allocations they produce are EF or EF1 in every time step during their execution.  \nIf we follow the classic model of online algorithm","cbCailCICG9UZ6Ln","https://ap.wps.com/l/cbCailCICG9UZ6Ln","pdf",676982,4,1,26,"English","en",105,"# Abstract\n# Introduction\n## Our Contribution","[{\"question\":\"How is the online fair division problem modeled in the paper?\",\"answer\":\"Items arrive one by one over time, and when an item arrives each agent gives it a value that can be positive, negative, or zero. The item must then be allocated irrevocably before the next item arrives.\"},{\"question\":\"What fairness notions does the paper focus on?\",\"answer\":\"It focuses on envy-freeness (EF) and a relaxation called envy-freeness up to one item (EF1), aiming for guarantees that hold at each time step.\"},{\"question\":\"Why are reordering buffers introduced, and what do they enable?\",\"answer\":\"Buffers let the algorithm store a limited number of arriving items and decide later how to allocate them. This added flexibility helps overcome the scarcity of positive results in the standard fully online model and supports strong EF1/EF guarantees with bounded buffer sizes.\"}]",1784188878,66,{"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},"online-fair-division-meets-reordering-buffers","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/online-fair-division-meets-reordering-buffers/83565/",{"url":52,"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},"How is the online fair division problem modeled in the paper?","Question",{"text":75,"@type":76},"Items arrive one by one over time, and when an item arrives each agent gives it a value that can be positive, negative, or zero. The item must then be allocated irrevocably before the next item arrives.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What fairness notions does the paper focus on?",{"text":80,"@type":76},"It focuses on envy-freeness (EF) and a relaxation called envy-freeness up to one item (EF1), aiming for guarantees that hold at each time step.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are reordering buffers introduced, and what do they enable?",{"text":84,"@type":76},"Buffers let the algorithm store a limited number of arriving items and decide later how to allocate them. This added flexibility helps overcome the scarcity of positive results in the standard fully online model and supports strong EF1/EF guarantees with bounded buffer sizes.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"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":20,"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"]