[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-141528-105":59,"doc-detail-141528-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","a-complexity-dichotomy-for-generalized-rainbow-matchings-based-on-color-classes","A Complexity Dichotomy for Generalized Rainbow Matchings - Based on Color Classes","","Given an edge-colored graph, the Maximum Rainbow Matching problem seeks a matching with maximum cardinality that uses at most one edge from each color. The work establishes a complexity dichotomy based on the structure of color classes: a polynomial-time algorithm exists when almost every color class forms a complete multipartite graph, while the problem is NP-hard otherwise. NP-hardness is proved even when each color class is restricted to small specific subgraphs on four vertices, then extended to cases beyond complete multipartite graphs. Positive results reduce the task to computing a maximum (l, u)-matching and give an efficient algorithm when all color classes are complete multipartite.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/a-complexity-dichotomy-for-generalized-rainbow-matchings-based-on-color-classes/141528/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/a-complexity-dichotomy-for-generalized-rainbow-matchings-based-on-color-classes/141528.png","ImageObject",300,407,{"name":92,"@type":93},"Jake","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-19","2026-08-25",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the Maximum Rainbow Matching problem require?","Question",{"text":112,"@type":113},"It asks for a matching of maximum size in an edge-colored graph such that edges used have pairwise distinct colors, i.e., at most one edge from each color class.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is the paper’s complexity dichotomy characterized?",{"text":117,"@type":113},"It states that the problem is polynomial-time solvable when almost every color class is a complete multipartite graph, and NP-hard otherwise.",{"name":119,"@type":110,"acceptedAnswer":120},"How is NP-hardness proved for the dichotomy?",{"text":121,"@type":113},"The proof shows NP-hardness even when every color class is a subgraph on four vertices of certain limited types (size-2 matching, 4-vertex path, or paw), then generalizes to color classes that are not complete multipartite graphs.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},141528,1787658825,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},962084928904,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","arXiv :2604 .21025v1 [ cs .DM] 22 Apr 2026  \nA Complexity Dichotomy for Generalized Rainbow Matchings  \nBased on Color Classes  \nFelix Hommelsheim∗1, Pia Jehmlich2 , and Moritz M¨uhlenthaler3  \n1 University of Cologne, Germany, [hommelsheim@cs.uni-koeln.de](hommelsheim@cs.uni-koeln.de), ORCID:  \n0000-0003-4444-9793  \n2 Freie Universit¨at Berlin, Germany, [p.jehmlich@fu-berlin.de](p.jehmlich@fu-berlin.de)  \n3 G-SCOP, Universit´e Grenoble-Alpes, France, [moritz.muhlenthaler@grenoble-inp.fr](moritz.muhlenthaler@grenoble-inp.fr), ORCID: 0000-0002-2729-127X  \nAbstract  \nGiven an edge-colored graph, the Maximum Rainbow Matching problem asks for a maximumcardinality matching of the graph that contains at most one edge from each color. We provide the following complexity dichotomy for this problem based on the structure of the color classes:  \nMaximum Rainbow Matching admits a polynomial-time algorithm if almost every color class is a complete multipartite graph and it is NP-hard otherwise.  \nTo prove the NP-hardness-part of the dichotomy, we first show that the problem remains NPhard even if every color class is a subgraph on four vertices that is either a matching of size two, a path on four vertices or a paw. We then leverage this result to all color classes that are not complete multipartite graphs. For this purpose, we introduce color-closed graph classes, which seem to be an appropriate notion for obtaining complexity classifications for rainbow problems and maybe of independent interest. To prove the positive part of the dichotomy, we show that the problem essentially reduces to computing a maximum (l, u)-matching, where we heavily exploit that almost all color classes are complete multipartite graphs. In the case where all color classes are complete multipartite, we provide a polynomial-time algorithm that computes a maximum matching containing at most mi edges from each color class i.  \n∗ Corresponding author  \n1 Introduction  \nGiven a combinatorial optimization problem with a set of feasible solutions S ⊆ 2X , a rainbow version of the problem assigns a color to each element of the ground set X and asks for an optimal solution S ∈ Sunder the additional restriction that each color appears at most once. Rainbow versions of classical problems that admit efficient algorithms, such as Maximum Matching or s–t Connectivity, are known to be NP-hard, even if each color appears at most twice in the ground set [32, 37] . A notable exception is the rainbow version of the Spanning Tree problem, which can be represented as the intersection of a graphic matroid and a partition matroid, and therefore admits a polynomial-time algorithm [34] .  \nRainbow versions of combinatorial optimization problems naturally model requirements related to fairness and diversity that are important both in theory and in practice. For example, in social networks, where vertices represent individuals labeled by demographic groups (colors) and edges represent conflicts, a rainbow independent set is a conflict-free group of representatives with at most one member from each demographic. Furthermore, in communication networks, channels (edges) may be labeled by frequency bands, and one may want to construct a path that avoids reusing the same frequency band in order to reduce the risk of interference; this corresponds to the rainbow s-t Connectivity problem. Finally, we may consider planning activities that can be carried out in pairs. The edges of the graph represent participants compatibility and the colors represent the different activities. For each activity i, there is a budget: it is available at most mi times. The goal is to maximize the number of activities carried out under budget and compatibility constraints; this is a maximum rainbow matching if all mi = 1 .  \nFrom a theoretical perspective, rainbow problems are interesting because they connect several classical topics in combinatorics and algorithms. For instance, they generalize transversals of latin sq","cbCairdXbD2lFLAP","https://ap.wps.com/l/cbCairdXbD2lFLAP","pdf",530354,17,"English","# Abstract\n# Introduction","[{\"question\":\"What does the Maximum Rainbow Matching problem require?\",\"answer\":\"It asks for a matching of maximum size in an edge-colored graph such that edges used have pairwise distinct colors, i.e., at most one edge from each color class.\"},{\"question\":\"How is the paper’s complexity dichotomy characterized?\",\"answer\":\"It states that the problem is polynomial-time solvable when almost every color class is a complete multipartite graph, and NP-hard otherwise.\"},{\"question\":\"How is NP-hardness proved for the dichotomy?\",\"answer\":\"The proof shows NP-hardness even when every color class is a subgraph on four vertices of certain limited types (size-2 matching, 4-vertex path, or paw), then generalizes to color classes that are not complete multipartite graphs.\"}]","A Complexity Dichotomy for Generalized Rainbow Matchings - Based on Color Classes | PDF",43]