[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82023-en":3,"doc-seo-82023-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},82023,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Covering Points with Rectangular Boundaries","Geometric covering problems seek a small collection of shapes whose union covers a given point set, but boundary covering requires every point to lie on the boundary of a chosen object. The study introduces axis-parallel rectangle boundary covering in discrete and continuous forms. The discrete version, BCDAPR, selects rectangles from a given family and is W[1]-hard with respect to k. The continuous version, BCCAPR, allows free placement and is fixed-parameter tractable in 2O(klog k)·nO(1), with NP-completeness shown for related L-shapes.","arXiv :2607 .08 183v 1 [ cs .CG] 9 Jul 2026  \nCovering Points with Rectangular Boundaries  \nMadhumita Kundu \\# University of Bergen, Norway Daniel Lokshtanov \\#  \nUniversity of California Santa Barbara, USA Soumi Nandi \\#   \nThe Institute of Mathematical Sciences, India Saket Saurabh \\#   \nThe Institute of Mathematical Sciences, India University of Bergen, Norway  \nKushal Singanporia \\#  \nThe Institute of Mathematical Sciences, India  \n~~ Abstract ~~  \nGeometric covering problems ask for a small family of geometric objects whose union covers a given point set. We study the more restrictive boundary covering variant, where every point must lie on the boundary of a chosen object. Motivated by the framework of Langerman and Morin [Discret. Comput. Geom., 2005] for hyperspheres, we initiate the study of boundary covering by axis-parallel rectangles.  \nWe first consider the discrete setting, where rectangles must be selected from a given family. We define Boundary Covering with Discrete Axis-Parallel Rectangles (BCDAPR): given a point set P ⊆ R2 , a family R of axis-parallel rectangles, and an integer k, decide whether P can be covered by the boundaries of at most k rectangles from R. We prove that BCDAPR is W[1]-hard parameterized by k.  \nWe then study the continuous variant, Boundary Covering with Continuous Axis-Parallel Rectangles (BCCAPR), where rectangles may be placed freely. Given P ⊆ R2 and k, the goal is to decide whether P can be covered by the boundaries of at most k axis-parallel rectangles. In contrast to the discrete case, we show that BCCAPR is fixed-parameter tractable, with running time 2O (klog k) · nO(1), where n = |P | . Our algorithm relies on a structural analysis of how k rectangles interact with the point set, reducing BCCAPR to at most 2 O (klog k) instances of Distinct Domain Monotone 2-CSP, each solvable in polynomial time. On the hardness side, we prove NP-completeness for boundary covering by axis-aligned L-shapes and use this reduction to establish NP-completeness of BCCAPR.  \n2012 ACM Subject Classification Theory of computation → Fixed parameter tractability; Theory of computation → Computational geometry; Theory of computation → Problems, reductions and completeness  \nKeywords and phrases Geometric Covering, Axis-parallel Rectangles, W[1] and NP Hardness, Fixed Parameter Tractability, CSP  \nDigital Object Identifier 10.4230/LIPIcs.CVIT.2016.23  \n© Jane Open Access and Joan R. Public;  \nlicensed under Creative Commons License CC-BY 4.0 42nd Conference on Very Important Topics (CVIT 2016) .  \nEditors: John Q. Open and Joan R. Access; Article No. 23; pp. 23:1–23:44  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n23:2 Covering Points with Rectangular Boundaries  \nContents  \n1 Introduction 3  \n1. 1 Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n1.2 Related work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5  \n2 Technical Overview of Our Results 5  \n2.1 Technical overview for W[1]-hardness for BCDAPR .............. 5  \n2.2 Overview of NP-hardness results . . . . . . . . . . . . . . . . . . . . . . . . . 7  \n2.3 Overview of an FPT algorithm for BCCAPR .................. 7  \n3 Notations and Preliminaries 9  \n3. 1 Definition of 3-Regular 2-CSP . . . . . . . . . . . . . . . . . . . . . . . . . 10  \n4 W[1]-hardness for BCDAPR 11  \n4. 1 Reduction from 3-Regular 2-CSP . . . . . . . . . . . . . . . . . . . . . . . 11  \n4.2 Correctness of the Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 13  \n5 NP-hardness for L-Shape Cover and BCCAPR 15  \n5. 1 Reduction to LSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15  \n5.2 Forcing lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17  \n5.3 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17  \n5.4 Orientations and re","cbCaiky7mQSTygqp","https://ap.wps.com/l/cbCaiky7mQSTygqp","pdf",1308246,5,1,44,"English","en",105,"# Introduction\n## Our Results\n## Related work\n# Technical Overview of Our Results\n## W[1]-hardness for BCDAPR\n## NP-hardness results\n## FPT algorithm for BCCAPR\n# Notations and Preliminaries\n# W[1]-hardness for BCDAPR\n## Reduction from 3-Regular 2-CSP\n## Correctness of the Reduction\n# NP-hardness for L-Shape Cover and BCCAPR\n## Reduction to LSC\n## Forcing lemmas\n## Correctness\n## Orientations and restriction\n# An FPT algorithm for BCCAPR","[{\"question\":\"What is the boundary covering problem studied in the paper?\",\"answer\":\"The paper studies boundary covering, where every input point must lie on the boundary of at least one chosen geometric object. This restricts coverage to boundaries rather than allowing points to be covered by shape interiors.\"},{\"question\":\"How do the discrete and continuous variants differ?\",\"answer\":\"In the discrete variant (BCDAPR), rectangles must be chosen from a given family. In the continuous variant (BCCAPR), rectangles can be placed freely in the plane.\"},{\"question\":\"What are the main complexity results for BCDAPR and BCCAPR?\",\"answer\":\"BCDAPR is W[1]-hard when parameterized by k. In contrast, BCCAPR is fixed-parameter tractable with running time 2O(klog k)·nO(1), and NP-completeness is also established for boundary covering by axis-aligned L-shapes, leading to NP-completeness of BCCAPR.\"}]","Covering Points with Rectangular Boundaries | PDF",1784177647,111,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"covering-points-with-rectangular-boundaries","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/covering-points-with-rectangular-boundaries/82023/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What is the boundary covering problem studied in the paper?","Question",{"text":77,"@type":78},"The paper studies boundary covering, where every input point must lie on the boundary of at least one chosen geometric object. This restricts coverage to boundaries rather than allowing points to be covered by shape interiors.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How do the discrete and continuous variants differ?",{"text":82,"@type":78},"In the discrete variant (BCDAPR), rectangles must be chosen from a given family. In the continuous variant (BCCAPR), rectangles can be placed freely in the plane.",{"name":84,"@type":75,"acceptedAnswer":85},"What are the main complexity results for BCDAPR and BCCAPR?",{"text":86,"@type":78},"BCDAPR is W[1]-hard when parameterized by k. In contrast, BCCAPR is fixed-parameter tractable with running time 2O(klog k)·nO(1), and NP-completeness is also established for boundary covering by axis-aligned L-shapes, leading to NP-completeness of BCCAPR.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":20,"slug":139},19,"General","general"]