[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86462-en":3,"doc-seo-86462-105":30,"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":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":13,"seo_description":14,"update_tm":28,"read_time":29},86462,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Characterization of Bichromatic Maximum-Sum Matchings and Matching Equilibrium","Study focuses on maximum-sum red-blue matchings and matching equilibrium for finite planar point sets. For a red-blue perfect matching M, the directed red-cycle gain is defined as the change in total weight under cyclic shifts of blue partners. M is maximum-sum exactly when every directed red cycle has nonpositive gain, with a geometric sufficient optimality condition from cyclic intersections of distance-difference regions. Balanced matchings and equilibrium are characterized via vanishing cycle gains, additive distance-matrix structure, and common level-set conditions, yielding orthogonality and hyperbolic level-set descriptions in squared and Euclidean settings.","arXiv :2607 . 10070v1 [math .CO] 11 Jul 2026  \nCharacterization of bichromatic maximum-sum matchings of points  \nand matching equilibrium  \nOscar Chac´on-Rivera∗  \nJuly 14, 2026  \nAbstract  \nWe study maximum-sum red-blue matchings and matching equilibrium for finite planar point sets. For a red-blue perfect matching M = { (ai , bi ) : 1 ≤ i ≤ n}, we define the gain of a directed red cycle as the change in total weight produced by cyclically shifting the corresponding blue partners. We prove that M is maximum-sum if and only if every directed red cycle has nonpositive gain, and we derive a geometric sufficient condition for optimality from cyclic intersections of distance-difference regions. We then characterize balanced matchings, in which all red-blue perfect matchings have the same total weight. Equilibrium is shown to be equivalent to vanishing cycle gains, to an additive form of the distance matrix, and to a common level-set condition for distance-difference functions. In the squared Euclidean case this yields an orthogonality classification, while in the Euclidean case it yields a hyperbolic level-set description anda collinear-separation classification in the nondegenerate setting.  \n1 Introduction  \nLet R and B be two point sets in the Euclidean plane with |R| = |B| . The points in R are called red points, and those in B are called blue points. A matching M of R ∪ B is a partition of R ∪ B into n pairs such that each pair consists of a red point and a blue point. A point p ∈ R and a point q ∈ B are matched by M if and only if the (unordered) pair (p, q) is in the matching M.  \nGiven a metric or a semi-metric function d : R2 × R2 → R≥0, we say that a matching M is max-sum if it maximizes P(p,q)∈Md(p, q) among all matchings of R and B. Recall that a function is a semi-metric if it satisfies all the properties of a metric function except for the triangle inequality.  \nChac´on-Rivera and P´erez-Lantero [6] characterized a maximum-sum matching in terms of Hsets and h-sets defined as  \nH (p, q) ={x ∈ R2 : d(p, q′) − d(q, q′) ≤ d (p, x) − d(q, x)} ,  \nand  \n h (p, q) ={x ∈ R2 : d (p, x) − d(q, x) ≤ d(p,p′) − d(q, p′)} ,∗ Pontificia Universidad Cat´olica de Chile, Facultad de Matem´aticas, Chile. opchacon@mat .uc .cl.  \nwhere p, q are red points, and p′, q′ are blue points, and {(p,p′),(q, q′)} is a maximum-sum matching of those four points, that is, M is a 2-local max-sum matching as defined by Biniaz et al. [5] . This characterization proved useful in simplifying the proof of the common intersection property of disks established by Huemer et al. [10] .  \nIn this article, we consider max-sum matchings between planar colored point sets R and B with |R| = |B| = n. In what follows, we generalize the characterization proposed by Chac´on-Rivera and P´erez-Lantero [6] to matchings of n red points and n blue points, and then use this characterization to study equilibrium phenomena in red-blue matchings.  \n1.1 Related work and motivation  \nGeometric matching problems often combine an optimization condition with intersection properties of the objects induced by the selected edges. In the setting of planar point sets, a matched pair naturally induces the disk having the segment joining the two matched points as diameter. Huemer et al. [10] proved that if R and B are finite point sets in the plane with |R| = |B|, and M is ared-blue perfect matching maximizing the total squared Euclidean length of its edges, then all diametral disks induced by the edges of M have a nonempty common intersection. This result showed that a global optimality condition on a matching may force a strong geometric piercing property.  \nThe analogous question for the ordinary Euclidean distance is more subtle. Bereg et al. [4] showed that, in the bichromatic Euclidean setting, the disks induced by a maximum-sum matching need not have a common point, although they satisfy weaker intersection properties. In contrast, they proved that for a set of 2n uncolored points ","cbCaisLublCJ6xQX","https://ap.wps.com/l/cbCaisLublCJ6xQX","pdf",358115,4,1,17,"English","en",105,"# Abstract\n# Introduction\n## Related work and motivation\n## Problem setup and preliminaries\n# Maximum-sum matchings and cycle gains\n# Geometric optimality conditions\n# Balanced matchings and equilibrium characterizations\n# Special cases: squared Euclidean and Euclidean","[{\"question\":\"How is maximum-sum optimality characterized for red-blue perfect matchings?\",\"answer\":\"A red-blue perfect matching is maximum-sum if and only if every directed red cycle has nonpositive gain, where gain measures the change in total weight from cyclically shifting blue partners.\"},{\"question\":\"What geometric condition is used to obtain a sufficient criterion for optimality?\",\"answer\":\"Optimality follows from cyclic intersections of distance-difference regions, which provide a geometric sufficient condition derived from the cycle-based gain framework.\"},{\"question\":\"What does matching equilibrium mean, and how is it related to cycle gains?\",\"answer\":\"Matching equilibrium is equivalent to vanishing cycle gains; it also corresponds to an additive form of the distance matrix and to a common level-set condition for distance-difference functions.\"}]",1784211872,43,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"characterization-of-bichromatic-maximum-sum-matchings-and-matching-equilibrium","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/characterization-of-bichromatic-maximum-sum-matchings-and-matching-equilibrium/86462/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","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},"How is maximum-sum optimality characterized for red-blue perfect matchings?","Question",{"text":75,"@type":76},"A red-blue perfect matching is maximum-sum if and only if every directed red cycle has nonpositive gain, where gain measures the change in total weight from cyclically shifting blue partners.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What geometric condition is used to obtain a sufficient criterion for optimality?",{"text":80,"@type":76},"Optimality follows from cyclic intersections of distance-difference regions, which provide a geometric sufficient condition derived from the cycle-based gain framework.",{"name":82,"@type":73,"acceptedAnswer":83},"What does matching equilibrium mean, and how is it related to cycle gains?",{"text":84,"@type":76},"Matching equilibrium is equivalent to vanishing cycle gains; 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