[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84635-en":3,"doc-seo-84635-105":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},84635,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Real-weighted Diameter and Eccentricity of Minor-free and Bounded VC-dimension Graphs in Truly Subquadratic Time","First truly subquadratic-time algorithm computes diameter and eccentricity in real-weighted directed graphs under constant-distance VC-dimension together with strongly sublinear-sized balanced separators. The runtime achieves O(n2−1/(2h−2) polylog(n)) for real-weighted Kh-minor-free digraphs. Prior work achieved subquadratic diameter mainly for real-weighted planar graphs or for minor-free graphs only in unweighted/integer-weight regimes. This paper introduces a randomized search-to-decision reduction to enable real-weight handling and shows VC-dimension remains algorithmically powerful in that setting.","arXiv :2607 .0 1926v 1 [ cs .DS] 2 Jul 2026  \nReal-weighted Diameter and Eccentricity of Minor-free and Bounded VC-dimension Graphs in Truly Subquadratic Time  \nDa Wei Zheng*  \nJuly 3, 2026  \nAbstract  \nWe present the first truly subquadratic time algorithm to compute diameter and eccentricity in real-weighted directed graphs with constant distance VC-dimension and strongly sublinear-sized balanced separators. This runs in O(n2−1/(2h−2)polylog(n)) time for real-weighted Kh-minor-free digraphs.  \nPrior to this work, truly subquadratic time computation of diameter was only known for real-weighted planar graphs, while extensions to broader classes like minor-free graphs were restricted to unweighted settings. In particular, existing algorithms that use VC-dimension [Ducoffe, Habib, Viennot; SICOMP 2022][Le, Wulff-Nilsen; SODA 2024][Chan, Chang, Gao, Le, KisfaludiBak, Zheng; FOCS 2025] work with small integer weights, but do not naturally generalize to real weights. We overcome this barrier by introducing a randomized search-to-decision reduction, demonstrating that VC-dimension is a sufficiently powerful tool in the real-weighted regime.  \n1 Introduction  \nFor a sparse graph G with n vertices and O (n) non-negative real-weighted edges, it is possible to  \ncomthepgutrapehtheHoiametever, erevoe Gin e(n2igh)tteimd ue1nby runndirectedingrg Daph,kstracom’s aputligongrthhemerom everact diameytevrrtnetxruofly  \nsubquadratic time is impossible assuming the Strong Exponential Time Hypothesis (SETH) [RV13 ] . This negative result has motivated the study of diameter in structured graph classes where truly subquadratic-time diameter is achievable, such as bounded-treewidth graphs [CK09, BHM20 ], planar graphs [Cab19, GKM+21], bounded genus graphs [KPPS25 ], Kh-minor-free graphs [DHV22, LW24, KZ25 ], and geometric intersection graphs [DKP24, CCG+25] .  \nTwo major paradigms have emerged for breaking this quadratic barrier for structured graph classes. The first is abstract Voronoi diagrams for planar graphs that were pioneered by Cabello [Cab19]  \natWiiemloeenKirilye wWiluaaatmacicptr-mveorae5/rr3aam) loyelmrypoenlgrKreaaappl-sneti,haMghtotensa,inShndri(,11/6anctedd)  \nsettings, the approach strongly relies on planarity and is likely only generalizable to bounded-genus graphs.  \nThe second paradigm revolves around VC-dimension. Ducoffe, Habib, and Viennot [DHV22 ] demonstrated that it is possible to get a truly subquadratic-time algorithm in undirected unweighted Kh-minor-free graph with distance VC-dimension, the VC-dimension of neighborhood balls centered  \n*This project has received funding from the Austrian Science Fund (FWF) grant DOI 10.55776/I5982 . For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.  \n1Throughout this paper, we will use (·) to suppress polylogarithmic factors of n.  \nGraph class  \nRuntime  \nReal-weighted?  \nPlanar  \nO (n11/6)  \n(n5/3)  \nY  \nY  \n[Cab19][GKM+21]  \nKh-minor-free  \n(((11////((2O32((h222)))))  \nN  \nN  \nN  \nN  \n[DHV22 ][LW24 ]  \n[KZ25 ][CCG+25]  \n(n2−1/(2h−2))  Y Corollary 2   \nO (nβ)-separator g.d. VC-dim d  \n2(−n2(β/()/22dO) )(d))  \nN  \nN  \n[DHV22 ][CCG+25]  \n(n2−(1−β )/d )  Y Theorem 1   \ng.d. VC-dim d O (mn1−1/(2d)) N [CCG+25]  \ne  \nTable 1: Summary of runtime guarantees for diameter and eccentricity algorithms in directed graphs with n vertices and m edges. Generalized distance VC-dimension is abbreviated as “g.d. VC-dim”. All bounds are for directed graphs. Note that all graphs besides g.d. VC-dim d graphs are sparse and have m = O (n) .  \nat every vertex of the graph. Le and Wulff-Nilsen [LW24 ] generalized a different bounded VCdimension set system –distance profiles – studied by Li and Parter [LP19 ] in planar graphs to obtain results for directed Kh-minor-free graphs. Bounds on both distance VC-dimension and distance profiles were unified by Karczmarz and Zheng [KZ25 ] with generalized distance VC-","cbCaikyvNyaWsc9D","https://ap.wps.com/l/cbCaikyvNyaWsc9D","pdf",218795,1,12,"English","en",105,"# Abstract\n# Introduction\n## Our Contribution","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper addresses computing graph diameter and vertex eccentricities in real-weighted directed graphs efficiently, specifically in truly subquadratic time under structural constraints.\"},{\"question\":\"What are the key assumptions enabling the algorithm?\",\"answer\":\"The approach relies on constant distance VC-dimension and strongly sublinear-sized balanced separators, and targets monotone graph families such as real-weighted Kh-minor-free digraphs.\"},{\"question\":\"How does the paper overcome the difficulty of real edge weights?\",\"answer\":\"It introduces a randomized search-to-decision reduction, showing that VC-dimension tools can be adapted to the real-weighted regime where earlier algorithmic machinery did not generalize 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problem does the paper address?","Question",{"text":75,"@type":76},"The paper addresses computing graph diameter and vertex eccentricities in real-weighted directed graphs efficiently, specifically in truly subquadratic time under structural constraints.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are the key assumptions enabling the algorithm?",{"text":80,"@type":76},"The approach relies on constant distance VC-dimension and strongly sublinear-sized balanced separators, and targets monotone graph families such as real-weighted Kh-minor-free digraphs.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper overcome the difficulty of real edge weights?",{"text":84,"@type":76},"It introduces a randomized search-to-decision reduction, showing that VC-dimension tools can be adapted to the real-weighted regime where earlier algorithmic machinery did not generalize 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