[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82569-en":3,"doc-seo-82569-105":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},82569,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Tighter bounds for weighted and unweighted shortest cycle approximation","Studies approximating the length of a graph’s shortest cycle, i.e., its girth, with fast algorithms and explicit time–approximation trade-offs. For unweighted n-node graphs, prior work gives an O(n1+2/k) algorithm achieving a (2k/3)-approximation for every integer k≥2. This paper extends the same trade-off to weighted undirected graphs with non-negative real edge weights, achieving a 2k/3-approximation in O(m+n1+2/k) time, improving earlier partial results. It also establishes fine-grained lower bounds for girth approximation and related problems in unweighted graphs.","arXiv :2607 .00938v 1 [ cs .DS] 1 Jul 2026  \nTighter bounds for weighted and unweighted shortest cycle  \napproximation  \nAvi Kadria and Liam Roditty and Virginia Vassilevska Williams  \nAbstract  \nWe study the problem of approximating the length of a shortest cycle in a given graph, known as the girth of the graph. The state-of-the-art approximation algorithms for unweighted graphs by Kadria et al. [SODA’22] and Rodit˜ty and Trabelsi [arXiv’25] achieve the following  \ntrade-off: for every integer k ≥ 2, there is an O(n1+2/k) time algorithm that achieves a (2k/3)-approximation for the girth in unweighted n-node graphs. The first result of this paper is to achieve the same trade-off for m-edge, n-no˜de graphs with non-negative real edge weights:  \na 2k/3-approximation algorithm running in O (m + n1+2/k) time. The dependence on m is unavoidable in weighted graphs. Our result improves on the work of Kadria et al. [SODA’23] and Ducoffe [ICALP’19 and SIDMA’21], who were only able to achieve such a trade-off for some values of k. We also prove new fine-grained lower bounds for girth approximation and related problems in unweighted graphs.  \n1 Introduction  \nThe length of a shortest cycle in a graph, known as the girth and denoted by g, is a key parameter often used to shed light on the structure of graph problems (e.g., [LW97, OPT01, HW16]) . The problem of computing a shortest cycle and girth in an undirected graph is a fundamental problem studied extensively for decades, both in unweighted graphs (e.g. ,[IR78, AYZ97, YZ97, LL09, RV12, KRS+22]), and weighted graphs (e.g., [LL09, RV11, RT13, Duc21, KRS+22, KRS+23]) .  \nComputing the exact value of the girth is computationally expensive: in weighted graphs, it is known to be equivalent to the All-Pairs Shortest Paths (APSP) problem, and in unweighted graphs, it is known that any fast algorithm requires Boolean Matrix Multiplication (BMM) [VW18] . Because exact computation is deemed prohibitive, fast approximation algorithms have been extensively studied. A (multiplicative) c-approximation algorithm outputs a cycle of length gˆ ≤ c · g whenever g is the girth of the underlying graph. The factor c, which is the largest ratio ~~gˆ~~g that the algorithm achieves, is called the stretch of the algorithm.  \nIn unweighted graphs, Itai and Rodeh [IR78] were the first to consider girth approximation, and presented ˜an O (n2 ) time algorithm that returns g ≤ gˆ ≤ 2⌈g/2⌉ . Lingas and Lundell [LL09]  \npresented an O (n1.5 ) time algorithm 1 that returns g ≤ gˆ ≤ 4⌈g/2⌉ . Later, Kadria, Roditty, Sidford, Vassilevska Williams, and Zwick [KRS+22] generalized these results and obtained the following trade-off. For any even integer k ≥ 2, there is an O(n1+2/k ) time algorithm that returns g ≤ gˆ ≤ k·⌈g/2⌉ . Recently, Roditty and Trabelsi [RT25] obtained the same trade-off for odd values of k. The corresponding stretch is at most ~~k~~2 when g is even, and at most ~~k~~2 ·(1+ ~~1~~g) when g is odd. If g = 3, that is, there is a triangle in the graph, then ~~k~~2 · (1 + ~~1~~g) = 2k/3, which is the stretch in the worst case. We can summarize the state of the art for unweighted graphs as:  \nFor every integer k ≥ 2, there is an ˜O(n1+ ~~ 2~~k ) time 2k/3-approximation algorithm for the girth of  \nunweighted graphs.  \nA central and extensively studied question in graph algorithms is whether the running timeapproximation trade-off established for unweighted graphs can be matched in the case of weighted undirected graphs with non-negative real edge weights. Therefore, the following problem is natural.  \nProblem 1 . Is it possible, for every integer k ≥ 2, to obtain an ˜O(m + n 1+ ~~ 2~~k ) time algorithm with ~~2k~~3-stretch in weighted undirected graphs with non-negative real edge weights?  \n(We allow an additive m in the running time above since in weighted graphs one needs to read the input even for an approximation2 .)  \nRoditty and Tov [RT13] almost solved Problem 1 for k = 2 and presented an ˜O( ~~1~~εn2","cbCaivweu7kf2Wpz","https://ap.wps.com/l/cbCaivweu7kf2Wpz","pdf",699904,1,41,"English","en",105,"# Introduction\n## Girth approximation background and trade-offs\n## Problem 1: matching unweighted and weighted trade-offs\n## Main theorem and improvements","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper addresses approximating the length of a graph’s shortest cycle, known as the girth.\"},{\"question\":\"What approximation and runtime does the paper achieve for weighted graphs?\",\"answer\":\"For weighted undirected graphs with non-negative real edge weights and odd k≥3, it provides an algorithm running in ÓO(m+n1+2/k) time that outputs an estimate ĝ with g ≤ ĝ ≤ (2k/3)g.\"},{\"question\":\"How does the paper compare to earlier work?\",\"answer\":\"It improves prior trade-off results for real weighted graphs, which previously worked only for some k values; the paper also tightens results for the k=3 case under real weights. It further adds fine-grained lower bounds for girth approximation in unweighted graphs.\"}]",1784181579,103,{"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},"tighter-bounds-for-weighted-and-unweighted-shortest-cycle-approximation","",{"@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/tighter-bounds-for-weighted-and-unweighted-shortest-cycle-approximation/82569/",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 problem does the paper address?","Question",{"text":75,"@type":76},"The paper addresses approximating the length of a graph’s shortest cycle, known as the girth.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What approximation and runtime does the paper achieve for weighted graphs?",{"text":80,"@type":76},"For weighted undirected graphs with non-negative real edge weights and odd k≥3, it provides an algorithm running in ÓO(m+n1+2/k) time that outputs an estimate ĝ with g ≤ ĝ ≤ (2k/3)g.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper compare to earlier work?",{"text":84,"@type":76},"It improves prior trade-off results for real weighted graphs, which previously worked only for some k values; the paper also tightens results for the k=3 case under real weights. 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