[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83793-en":3,"doc-seo-83793-105":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},83793,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Beyond Trees: The Weighted Center Problem on Gromov Hyperbolic Graphs","The Weighted Center problem on graphs takes a graph G=(V,E) and a nonnegative multiplicative weight profile π:V→R≥0 and seeks a vertex c minimizing the maximum weighted distance max{π(v)dG(c,v):v∈V}. The classic Center problem is the special case π(v)=1. The work studies how Gromov hyperbolicity, combined with other metric/geometric graph properties, enables exact and approximate almost-linear time algorithms. Results provide almost optimal algorithms for chordal graphs, distance-hereditary graphs (O(m)), and dually chordal and chordal bipartite graphs (O(mlog n)).","arXiv :2607 .04287v 1 [ cs .DS] 5 Jul 2026  \nBEYOND TREES: THE WEIGHTED CENTER PROBLEM ON GROMOV  \nHYPERBOLIC GRAPHS  \nGUILLAUME DUCOFFE  \nAbstract . The Weighted Center problem takes as its input a graph G = (V, E) together with a profile π such that every vertex v is mapped to some nonnegative multiplicative weight π (v) . Its output must be some vertex c minimizing max{π(v)dG (c, v) : v ∈ V } . The classic Center problem corresponds to the case where π (v) = 1 for every vertex v. In the literature, various almost linear-time algorithms have been proposed for the Center problem on some well-structured classes of graphs. By contrast, similarly efficient algorithms for the Weighted Center problem have been scarce. We investigate how the Gromov hyperbolicity, alone or in combination with other metric and geometric properties on graphs, can be used in the design of exact and approximate almost linear-time algorithms for the Weighted Center problem. In particular, we derive almost optimal algorithms for the following well-studied classes of graphs:  \nchordal graphs, distance-hereditary graphs (both in O (m) time), dually chordal graphs and chordal bipartite graphs (both in O (mlog n) time) .  \n1. Introduction  \nProblem considered. We refer to [6] for basics in Graph Theory. Unless stated otherwise, graphs considered throughout the paper will be simple, undirected, unweighted and connected. As usual, let dG denote the shortest-path metric of a graph G = (V, E) , i.e. , for any vertices u and v their distance dG (u, v) is equal to the minimum number of edges on a (u, v)-path. The following facility location problem on graphs is considered:  \nWeighted Center  \nInput: A graph G = (V, E); a profile π : V →7 R≥0 .  \nOutput: A vertex c ∈ V s.t. rπ(c) = max{π(v)dG (c, v) : v ∈ V } is minimized.  \nRoughly, π (v) represents the importance for vertex v to be close to the new facility. For example, π (v) can be the expected material flux between v and a new factory, the number of children in a building at v who will attend to a new neighborhood school, etc. Furthermore, if we set π (v′) = 0 for some vertices v′, then we can account for the plausible situation where the placement of a new facility is only relevant for a subset of the vertices. Note that we cannot simply remove the zero-weight vertices for it could change the distances in the graph.  \nIn line with existing literature [36], the function rπ is called a radius function. The solution vertex c is called a center. The Weighted Center problem can be defined more generally for any metric space. In particular, a linear-time algorithm for the Weighted Center problem on Euclidean spaces of constant dimension was presented in [31] . For graphs, the existing literature is mostly focused on the following restricted variant:  \nCenter  \nInput: A graph G = (V, E) .  \nOutput: A vertex c ∈ V s.t. ecc(c) = max{dG (c, v) : v ∈ V } is minimized.  \nThe radius function ecc is also called the eccentricity function of G. More generally, if π(v) ∈{0, 1} for every vertex v, then π is called a binary profile, and rπ is called a binary radius function. The special case of binary profiles on graphs has received some attention in the literature, see [10, 15, 22] . As a rule of thumb, most of the techniques for the Center problem can be also applied for binary profiles. However, such is not the case for the more general Weighted Center problem.  \n2 G. DUCOFFE  \nRelated work. The Weighted Center problem can be reduced to All-Pairs ShortestPaths. For n-vertex m-edge graphs, this straightforward reduction leads to an O (nm)-time algorithm. In [1], the Hitting Sets Conjecture was introduced to prove a conditional lower bound in Ω(n2−o(1)) for the Center problem on n-vertex graphs, even if the number of edges is at most in n 1+o(1) . Therefore, breaking the quadratic barrier for the Center problem, and even more so for the Weighted Center problem, is likely to require additional restrictions on the classes of grap","cbCaiastGGQ0DysN","https://ap.wps.com/l/cbCaiastGGQ0DysN","pdf",745541,1,25,"English","en",105,"# Introduction\n# Problem considered\n# Related work\n# G. Ducoffe","[{\"question\":\"What is the objective of the Weighted Center problem on a graph?\",\"answer\":\"Given a graph G and a weight profile π on vertices, the task is to find a vertex c that minimizes the maximum value of π(v)·dG(c,v) over all vertices v.\"},{\"question\":\"How does the classic Center problem relate to the Weighted Center problem?\",\"answer\":\"The classic Center problem corresponds to the special case where every vertex has weight π(v)=1, turning weighted distances into standard eccentricity.\"},{\"question\":\"Which graph classes receive almost linear-time algorithms, and what running times are claimed?\",\"answer\":\"The paper derives almost optimal algorithms for chordal graphs and distance-hereditary graphs in O(m) time, and for dually chordal graphs and chordal bipartite graphs in O(mlog n) time.\"}]",1784190443,63,{"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-trees-the-weighted-center-problem-on-gromov-hyperbolic-graphs","",{"@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-trees-the-weighted-center-problem-on-gromov-hyperbolic-graphs/83793/",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 objective of the Weighted Center problem on a graph?","Question",{"text":75,"@type":76},"Given a graph G and a weight profile π on vertices, the task is to find a vertex c that minimizes the maximum value of π(v)·dG(c,v) over all vertices v.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the classic Center problem relate to the Weighted Center problem?",{"text":80,"@type":76},"The classic Center problem corresponds to the special case where every vertex has weight π(v)=1, turning weighted distances into standard eccentricity.",{"name":82,"@type":73,"acceptedAnswer":83},"Which graph classes receive almost linear-time algorithms, and what running times are claimed?",{"text":84,"@type":76},"The paper derives almost optimal algorithms for chordal graphs and distance-hereditary graphs in O(m) time, and for dually chordal graphs and chordal bipartite graphs in O(mlog n) time.","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"]