[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82802-en":3,"doc-seo-82802-105":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},82802,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Beyond Monotone Delays for Multi-Level Aggregation","Online Multi-Level Aggregation Problem (MLAP) models time-varying requests on nodes of a weighted rooted tree. Each request must be served by choosing a rooted subtree containing its node, paying service cost equal to the selected subtree’s total weight, while also incurring a time-dependent penalty determined by the serving time. The goal minimizes combined service and penalty costs. Extending prior delay-function assumptions, the work allows arbitrary, non-monotone penalty functions and arbitrary tree depth, presenting a randomized polynomial-time algorithm with competitive guarantees and hardness results.","arXiv :2607 .043 17v 1 [ cs .DS] 5 Jul 2026  \nBeyond Monotone Delays for Multi-Level Aggregation  \nYossi Azar∗ Liad Iluz†  \nAbstract  \nIn the online Multi-Level Aggregation Problem (MLAP), requests arrive over time and are associated with nodes of a given weighted rooted tree of depth 􀀙 . Each request must eventually be served by performing a service. Serving a request consists of selecting a rooted subtree that contains the request’s node, incurring a service cost equal to the total weight of the selected subtree. To reduce service costs, multiple requests may be served simultaneously by selecting a single rooted subtree that spans all of them. In addition, each request is associated with a penalty function that specifies the cost incurred when the request is served at a particular time. The objective is to minimize the total cost, consisting of both service costs and penalty costs.  \nMost previous work on MLAP assumes monotone non-decreasing penalty functions, commonly referred to as delay functions. Only very recent results consider penalty functions that initially decrease and subsequently increase, and even then only for the special cases of depths 􀀙 = 1 and 􀀙 = 2, namely the Joint Replenishment Problem (JRP) .  \nIn this work, we extend previous results in two ways. First, we allow arbitrary penalty functions, which may decrease and increase multiple times. Second, we study the general MLAP with arbitrary tree depth 􀀙 under these arbitrary penalty functions. We present a randomized algorithm which is 􀀤 (􀀙 log 􀀽 log(􀀽􀀙􀀬))-competitive, where 􀀬 is the maximum service window among all penalty functions after normalizing the Lipschitz parameter of each penalty function to be 1 and normalizing the minimum positive edge weight incident to the root to be 1; and 􀀽 is the number of requests. We note that our algorithm runs in polynomial-time, and even for 􀀙 = 1 the problem admits hardness of approximation of Ω(log 􀀽) for polynomial time algorithms.  \nAs mentioned above, prior to our work even for trees of depth 􀀙 = 1, 2, non-monotone penalty functions have been studied only in special cases of functions that decrease and increase only once. In contrast, for such trees we obtain 􀀤 (log 􀀽 log(􀀽􀀬))-competitive algorithms for arbitrary non-monotone penalty functions.  \n1 Introduction  \nThe online Multi-Level Aggregation Problem (MLAP) is a general aggregation problem that captures the trade-off between waiting in order to batch requests, thereby sharing an expensive resource, and serving requests at an optimal time to reduce time-dependent penalties. Every request arrives at a node of a given weighted rooted tree T and must eventually be served. A service corresponds to selecting a rooted subtree T′ ⊆ T, where all requests located in T′ can be served simultaneously. The service cost or the cost of performing a service is defined as the total weight of the edges in T′. In addition, each request incurs a time-dependent penalty cost, which depends on the time at which the request is served. The objective is to minimize the total cost, defined as the sum of the service costs and the penalties incurred by all requests.  \nThe depth of the tree 􀀙, is a key structural parameter of the problem. When 􀀙 = 1, MLAP is equivalent to the TCP Acknowledgment Problem (TCP-AP), where the tree consists only of a root and a single child,  \n∗Department of Computer Science, Tel Aviv University, Israel. Email: [azar@tau.ac.il](azar@tau.ac.il)  \n†Department of Computer Science, Tel Aviv University, Israel. Email: [liadiluz@mail.tau.ac.il](liadiluz@mail.tau.ac.il), [liadiluz8@gmail.com](liadiluz8@gmail.com)  \nand the edge weight represents the cost of performing an acknowledgment (also called single-item JRP) . When 􀀙 = 2, MLAP is equivalent to the Joint Replenishment Problem (JRP), which models multiple items ordered with a fixed order cost (the cost of the root) and with item-specific costs (the cost of the second level) .  \nMost prior work on the various var","cbCaimkEXf15Esaq","https://ap.wps.com/l/cbCaimkEXf15Esaq","pdf",819307,1,27,"English","en",105,"# Abstract\n# Introduction\n## Problem Definition (MLAP)\n## Relation to TCP-AP and JRP\n## Motivation: Beyond Monotone Delay Functions","[{\"question\":\"What is the online Multi-Level Aggregation Problem (MLAP) in this work?\",\"answer\":\"Requests arrive over time at nodes of a weighted rooted tree. Each request must be served by selecting a rooted subtree containing its node, paying service cost as the subtree’s total weight plus a time-dependent penalty based on when service occurs.\"},{\"question\":\"How does this paper extend previous MLAP research?\",\"answer\":\"It allows arbitrary penalty functions that can decrease and increase multiple times, and it studies MLAP for general (not just constant) tree depth under these arbitrary penalties.\"},{\"question\":\"What performance guarantees and limitations does the paper provide?\",\"answer\":\"It presents a randomized polynomial-time algorithm that is competitively optimal up to factors involving the tree depth and the maximum service window of penalties. It also notes hardness of approximation of order Ω(log n) even when the tree depth is 1 for polynomial-time algorithms.\"}]",1784183037,68,{"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},"beyond-monotone-delays-for-multi-level-aggregation","",{"@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/beyond-monotone-delays-for-multi-level-aggregation/82802/",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 online Multi-Level Aggregation Problem (MLAP) in this work?","Question",{"text":75,"@type":76},"Requests arrive over time at nodes of a weighted rooted tree. Each request must be served by selecting a rooted subtree containing its node, paying service cost as the subtree’s total weight plus a time-dependent penalty based on when service occurs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does this paper extend previous MLAP research?",{"text":80,"@type":76},"It allows arbitrary penalty functions that can decrease and increase multiple times, and it studies MLAP for general (not just constant) tree depth under these arbitrary penalties.",{"name":82,"@type":73,"acceptedAnswer":83},"What performance guarantees and limitations does the paper provide?",{"text":84,"@type":76},"It presents a randomized polynomial-time algorithm that is competitively optimal up to factors involving the tree depth and the maximum service window of penalties. It also notes hardness of approximation of order Ω(log n) even when the tree depth is 1 for polynomial-time algorithms.","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},[93,97,101,105,110,115,120,123,128,131,135],{"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":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]