[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85296-en":3,"doc-seo-85296-105":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},85296,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Randomization Helps in Online Graph Exploration: Breaking the Deterministic Lower Bound on Cycles","Online graph exploration requires an agent to visit every vertex of an initially unknown weighted graph and return to its start, learning adjacency only at visited vertices. Building on the deterministic-only line of work, this study proves that randomization can strengthen competitive guarantees by focusing on cycles, a graph class capturing core exploration difficulties. The RandHeavyTest randomized algorithm achieves a competitive ratio of at most 1.315, separating it from the deterministic optimum (~1.366), with matching upper and new lower bounds.","arXiv :2607 . 1 1203v 1 [ cs .DS] 13 Jul 2026  \nRandomization Helps in Online Graph Exploration: Breaking the Deterministic Lower Bound on Cycles  \nJúlia Baligács∗ Jan Hązła† Lena Volk‡  \nAbstract  \nIn online graph exploration, introduced by Kalyanasundaram and Pruhs (1994), an agent must visit all vertices of an initially unknown weighted graph and return to its starting position, while the graph is revealed only locally at visited vertices. Although the problem has attracted considerable attention, previous work has focused exclusively on deterministic algorithms. Randomized strategies are often substantially harder to analyze because of a fundamental challenge inherent to exploration. In this work, we give the first positive result showing that randomization can improve competitive guarantees in online graph exploration. To this end, we focus on cycles, a simple graph class which nevertheless captures a key difficulty of online exploration.  \nOur main contribution is RandHeavyTest, a randomized algorithm for online exploration of cycles whose competitive ratio we prove to be at most 1.315. This establishes a strict separation from the deterministic setting, where the optimal competitive ratio is ≈ 1.366, and thus gives the first provable advantage of randomization in online graph exploration. A key step towards this result is a new, simplified optimal deterministic algorithm, HeavyTest, whose formulation naturally suggests the randomized variant. We complement our upper bounds with lower bounds of 1.115 for arbitrary randomized algorithms and 1.207 for the natural class of so-called forward-greedy algorithms, which includes RandHeavyTest.  \n1 Introduction  \nIn the classical online graph exploration problem, introduced by Kalyanasundaram and Pruhsin 1994 [KP94], a single agent must explore an undirected, connected graph G = (V, E) with nonnegative edge weights w : E → R≥0 . The agent starts at some vertex s ∈ V and initially has no knowledge of the graph. Whenever it visits a vertex for the first time, it learns the identifiers of all adjacent vertices together with the weights of the corresponding incident edges. The agent may then decide which known edge to traverse next. The cost of traversing an edge is simply its weight. The goal is to visit all vertices and return to s, subject to minimizing the total cost.  \nAs usual, the quality of an exploration algorithm is measured in terms of competitive analysis. For a deterministic algorithm Alg, let Alg (G, s) denote the total cost incurred by Alg on graph G when started at vertex s. Let Opt(G) denote the offline optimum cost, that is, the minimum length of a closed walk in G that visits every vertex. Note that Opt (G) is independent of the starting  \n∗ University of Oxford. Email: [jbaligacs@gmail.com](jbaligacs@gmail.com. Funded)[. Funded](jbaligacs@gmail.com. Funded) by the European Union through the European Research Council under the project BOBR (grant agreement No. 948057) during employment in Warsaw and under the project CCOO (grant agreement No. 101165139) during employment in Oxford. Views and opinions expressed are however those of the authors 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 be held responsible for them.  \n†AIMS Rwanda. Email: [jan.hazla@aims.ac.rw](jan.hazla@aims.ac.rw. Supported by the Alexander von Humboldt Foundation German)[. Supported by the Alexander von Humboldt Foundation German](jan.hazla@aims.ac.rw. Supported by the Alexander von Humboldt Foundation German)[ ](jan.hazla@aims.ac.rw. Supported by the Alexander von Humboldt Foundation German)research chair funding and the associated DAAD project No. 57761435.  \n‡Technische Universität Darmstadt. Email: [volk@mathematik.tu-darmstadt.de](volk@mathematik.tu-darmstadt.de).  \nvertex, whereas the cost of an online algorithm may depend on it. A deterministic algorithm Algis ρ-competitive if Al","cbCaihXP5CpVphkT","https://ap.wps.com/l/cbCaihXP5CpVphkT","pdf",535244,3,1,19,"English","en",105,"# Introduction\n## Problem setting and competitive analysis\n## Randomization in online graph exploration\n## Main results (cycles and competitive ratios)","[{\"question\":\"What is the online graph exploration problem studied in this document?\",\"answer\":\"An agent must explore an unknown connected weighted graph, learning neighbors only when a vertex is first visited, and then return to the starting vertex after visiting all vertices. The goal is to minimize total traversal cost.\"},{\"question\":\"What new randomized algorithm is proposed for exploring cycles online?\",\"answer\":\"The document introduces RandHeavyTest, a randomized algorithm tailored to online exploration of cycles.\"},{\"question\":\"How does randomization improve the competitive ratio compared with deterministic algorithms?\",\"answer\":\"RandHeavyTest is proven to have a competitive ratio at most 1.315, which strictly improves over the deterministic setting where the optimal ratio is approximately 1.366. The paper also provides lower bounds for randomized algorithms, including for forward-greedy variants.\"}]",1784202317,48,{"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},"randomization-helps-in-online-graph-exploration-breaking-the-deterministic-lower-bound-on-cycles","",{"@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/randomization-helps-in-online-graph-exploration-breaking-the-deterministic-lower-bound-on-cycles/85296/",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","2026-07-24","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 graph exploration problem studied in this document?","Question",{"text":75,"@type":76},"An agent must explore an unknown connected weighted graph, learning neighbors only when a vertex is first visited, and then return to the starting vertex after visiting all vertices. The goal is to minimize total traversal cost.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What new randomized algorithm is proposed for exploring cycles online?",{"text":80,"@type":76},"The document introduces RandHeavyTest, a randomized algorithm tailored to online exploration of cycles.",{"name":82,"@type":73,"acceptedAnswer":83},"How does randomization improve the competitive ratio compared with deterministic algorithms?",{"text":84,"@type":76},"RandHeavyTest is proven to have a competitive ratio at most 1.315, which strictly improves over the deterministic setting where the optimal ratio is approximately 1.366. The paper also provides lower bounds for randomized algorithms, including for forward-greedy variants.","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},[93,97,101,105,110,115,120,123,128,131,135],{"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":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":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":22,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},"General","general"]