[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85774-en":3,"doc-seo-85774-105":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},85774,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Decomposing a Simple Polygon with Geodesic Unit-Balls","The document studies covering and partitioning a simple polygon into pieces whose geodesic size is limited by either unit geodesic radius or unit geodesic diameter, measured using the ℓ2-metric. Exact solutions are unknown and the problem is NP-hard for polygons with holes. The work develops polynomial-time approximation algorithms: first factor-9 approximations for both covering and partitioning in the radius case, and a factor-15 approximation for diameter partition, improving prior 72-approximation results.","arXiv :2607 .09983v1 [ cs .CG] 10 Jul 2026  \nDecomposing a Simple Polygon with Geodesic Unit-Balls  \nReilly Browne \\# 􀀚  \nDepartment of Computer Science, Dartmouth College, Hanover, NH, USA Prahlad Narasimhan Kasthurirangan \\# 􀀚  \nDepartment of Applied Mathematics and Statistics, Stony Brook University, USA  \n~~ Abstract ~~  \nWe consider covering and partitioning a simple polygon into pieces which either have unit geodesic radius or unit geodesic diameter, using the ℓ2-metric for distances. There is no known method for finding an exact solution to these problems, even when the input size is constant, and the problem is known to be NP-hard in the case of polygons with holes. With this in mind, we instead devote our attention to developing simple approximation algorithms that run in polynomial time. For the radius problem, we present the first known approximation algorithms for both covering and partitioning, achieving a factor of 9 . For the diameter problem, we are only able to give a positive result for the partition version of the problem, where we improve upon a complicated 72-approximation from Abrahamsen and Rasmussen [2], achieving a simple 15-approximation.  \n2012 ACM Subject Classification Theory of computation → Computational geometry; Theory of computation → Packing and covering problems; Theory of computation → Approximation algorithms analysis  \nKeywords and phrases Covering, partitioning, polygon, k-center, constant factor approximation. Digital Object Identifier 10.4230/LIPIcs.ESA.2026.65  \nFunding Prahlad Narasimhan Kasthurirangan: Supported by the US Office of Naval Research under grant no. N00014-26-1-2088 .  \nAcknowledgements We would like to thank the anonymous reviewers of ESA 2026 for their valuable feedback.  \n 1  Introduction  \nA fundamental problem which appears in many domains is that of describing large regions asa series of smaller, easier to process pieces. Depending on the application, there are varying definitions of small which are relevant. In some settings, we want pieces which have some simple property to work with, such as convex pieces or star-shaped pieces [23, 27], while in others, we care more about physical size, such as whether a piece fits inside of a square [2, 3 , 1] or that this division is, in some sense, equitable [9, 25 , 5] . There is also the concern of whether the pieces are allowed to overlap, in which case we are finding a cover of the region, or whether the pieces must be (interior) disjoint, in which case we are looking for a partition and whether we are covering a continuous domain or a discrete set of points [15, 18, 21, 6] .  \nOur focus will be on two notions of geodesic smallness, specifically pieces which have unit geodesic radius or unit geodesic diameter (see Section 1.1 for formal definitions), and our region of interest will be a simple polygon P.  \nIn the radius problem, we are finding a set of center points such that every point in P is within unit geodesic distance of at least one center. This notion of “small” is natural for facility location applications and is equivalent to the dual of the famous k-center problem [33, 13 , 30 , 14 , 11]: In k-center, we have a fixed budget of k centers and want to instead minimize the maximum distance from a point to its nearest center. The problem of covering a  \n© Reilly Browne and Prahlad Narasimhan Kasthurirangan;  \nlicensed under Creative Commons License CC-BY 4.0 34th Annual European Symposium on Algorithms (ESA 2026) .  \nEditors: Philip Bille, Seth Pettie, and Sabine Storandt; Article No. 65; pp. 65:1–65:16  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n65:2 Decomposing a Simple Polygon with Geodesic Unit-Balls  \ndiscrete set of points in P with geodesic unit-balls admits a local search–based PTAS [28, 20], owing to the existence of planar supports [8] . In contrast, for covering (a continuum in) P , progress has been limited. Rab","cbCaihFIWMcvesY5","https://ap.wps.com/l/cbCaihFIWMcvesY5","pdf",873647,1,16,"English","en",105,"# Introduction\n## Geodesic notions and problem variants\n# Preliminaries\n## Distances and geodesic balls\n## Small geodesic radius and diameter pieces","[{\"question\":\"What does it mean to cover or partition a simple polygon with unit geodesic radius pieces?\",\"answer\":\"Covering selects pieces with geodesic radius at most 1 such that their union equals the polygon. Partitioning further requires interior-disjoint pieces while still covering the polygon.\"},{\"question\":\"Why are exact algorithms not pursued in this work?\",\"answer\":\"No exact method is known even for fixed input size, and the radius/partitioning problem becomes NP-hard for polygons with holes.\"},{\"question\":\"What approximation guarantees are provided for the radius and diameter variants?\",\"answer\":\"For the geodesic radius problem, the paper gives first known factor-9 approximation algorithms for both covering and partitioning. For the diameter problem, it provides a factor-15 approximation for the partition version, improving over a prior 72-approximation.\"}]",1784206178,40,{"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},"decomposing-a-simple-polygon-with-geodesic-unit-balls","",{"@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/decomposing-a-simple-polygon-with-geodesic-unit-balls/85774/",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 does it mean to cover or partition a simple polygon with unit geodesic radius pieces?","Question",{"text":75,"@type":76},"Covering selects pieces with geodesic radius at most 1 such that their union equals the polygon. Partitioning further requires interior-disjoint pieces while still covering the polygon.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are exact algorithms not pursued in this work?",{"text":80,"@type":76},"No exact method is known even for fixed input size, and the radius/partitioning problem becomes NP-hard for polygons with holes.",{"name":82,"@type":73,"acceptedAnswer":83},"What approximation guarantees are provided for the radius and diameter variants?",{"text":84,"@type":76},"For the geodesic radius problem, the paper gives first known factor-9 approximation algorithms for both covering and partitioning. 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