[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83197-en":3,"doc-seo-83197-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83197,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Stable Matchings with Minimum Utility Gap","Stable Matchings with Minimum Utility Gap introduces a fairness-oriented stable matching objective that seeks balanced utilities across agents. The framework supports many-to-many matchings and general utility functions over partner sets consistent with agents’ preferences. Two utility-comparison measures are studied: the gap between an agent’s maximum and minimum utility, and the ratio between them. A polynomial-time algorithm is provided for both versions using a rotation-poset representation and a chain property for rotations affecting each agent.","arXiv :2607 .07 160v 1 [ cs .GT] 8 Jul 2026  \nStable Matchings with Minimum Utility Gap  \nYao Sheng∗ Yu Yokoi†  \nAbstract  \nWe introduce the Stable Matching Problem with Minimum Utility Gap, which seeks astable matching in which the utilities received by individual agents are as balanced as possible. Our framework can handle many-to-many matchings and general utility functions on partner sets that are consistent with the agents’ preferences. We consider two measures for comparing agents’ utilities: the difference between the maximum and minimum utilities, and their ratio.  \nWe provide a polynomial-time algorithm for both versions. The algorithm exploits the rotation-poset representation of the set of stable matchings and, in particular, the fact that the rotations affecting each agent form a chain in this poset. To position our result, we also clarify its relation to existing frameworks: we show that our objectives are not captured by the recent minimum-cut representability framework, while identifying a special case that admits a submodular function minimization interpretation.  \n1 Introduction  \nSince the seminal work of Gale and Shapley [9], the theory of stable matching has grown into a rich research area, with deep mathematical structure and broad applications; see, e.g., [22, 24] . A matching in a two-sided market is stable if there is no unmatched pair of agents on opposite sides who can mutually benefit by deviating from their current assignments. The Gale–Shapley deferred-acceptance algorithm guarantees the existence of a stable matching. Throughout this paper, we denote the two sides of the market by S and L (students and laboratories), and call each element of S ∪ L an agent.  \nEven in the classical one-to-one setting, known as the stable marriage problem, the number of stable matchings can be exponential in the input size [12, 14] . The stable matching returned by the Gale–Shapley algorithm is optimal for one side, but pessimal for the other side [12, 21] . It is then natural to ask whether we can find a stable matching that is quantitatively optimal with respect to a given objective function.  \nIndeed, various stable matching optimization problems have been studied since the 1980s, especially for the marriage setting. The egalitarian stable matching problem asks for a stable matching M minimizing Pa∈S∪L ranka(M(a)), where ranka(M(a)) ∈ {1, 2 ,... } is the rank of the assigned partner of agent a; a smaller rank is better [12, 15] . The minimum-regret stable matching problem asks for a stable matching minimizing max a∈S∪L ranka(M(a)) [11, 12] . Both of these problems are tractable. More generally, Irving–Leather–Gusfield [15] showed that, for any weight function on acceptable pairs, a stable matching of minimum or maximum total weight  \n∗ Department of Mathematical and Computing Science, School of Computing, Institute of Science Tokyo,  \nTokyo 152-8552, Japan. Email: [sheng.y.2ab1@m.isct.ac.jp](sheng.y.2ab1@m.isct.ac.jp)  \n†Department of Mathematical and Computing Science, School of Computing, Institute of Science Tokyo,  \nTokyo 152-8552, Japan. Email: [yokoi@comp.isct.ac.jp](yokoi@comp.isct.ac.jp)  \ncan be found efficiently.1 This tractability is based on the distributive lattice structure of stable matchings and its compact representation by a rotation poset, which allows the problem tobe reduced to a minimum-cut problem [12] . Recent work of Faenza–Foussoul–He [6] gives a characterization of stable matching optimization problems, not necessarily linear, that can be represented as a minimum-cut problem.  \nOn the other hand, not all natural objective functions over stable matchings are tractable. For example, the sex-equal stable marriage problem asks for a stable matching M minimizing |Ps∈S ranks(M(s)) − P ℓ∈L rankℓ(M(ℓ))|, and is NP-hard [12, 20] . The balanced stable marriage problem asks for a stable matching minimizing the maximum of Ps∈S ranks(M(s)) and P ℓ∈L rankℓ(M(ℓ)), and is also NP-hard [8, 10] . These o","cbCaiqPO7Cu4JIgR","https://ap.wps.com/l/cbCaiqPO7Cu4JIgR","pdf",721871,5,1,20,"English","en",105,"# Introduction\n## Related stable matching optimization problems\n## Fairness objective based on individual utilities\n# Our Contribution\n## Difference version of the utility-gap objective\n## Ratio version of the utility-gap objective\n## Algorithmic solvability and connections","[{\"question\":\"什么是 Minimum Utility Gap 稳定匹配问题？\",\"answer\":\"该问题要求在所有稳定匹配中选择一个使得被关心代理的效用在个体之间尽可能均衡的匹配。衡量均衡性的方式包括“差值版本”和“比值版本”。\"},{\"question\":\"文中如何刻画效用并扩展到多对多匹配？\",\"answer\":\"每个代理在自身偏好约束下，对其被分配的伙伴集合定义一个效用函数；并引入容量使模型支持多对多匹配。效用函数需与偏好一致，即更优的伙伴集合对应更大的效用值。\"},{\"question\":\"差值版本与比值版本能否高效求解？\",\"answer\":\"文中给出结论：差值版本与比值版本都可以在同一算法框架下通过多项式时间高效求解，只需做少量修改即可。算法利用稳定匹配的 rotation-poset 表示，并使用与每个代理相关的旋转形成链这一性质。\"}]",1784185901,50,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"stable-matchings-with-minimum-utility-gap","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/stable-matchings-with-minimum-utility-gap/83197/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"什么是 Minimum Utility Gap 稳定匹配问题？","Question",{"text":76,"@type":77},"该问题要求在所有稳定匹配中选择一个使得被关心代理的效用在个体之间尽可能均衡的匹配。衡量均衡性的方式包括“差值版本”和“比值版本”。","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"文中如何刻画效用并扩展到多对多匹配？",{"text":81,"@type":77},"每个代理在自身偏好约束下，对其被分配的伙伴集合定义一个效用函数；并引入容量使模型支持多对多匹配。效用函数需与偏好一致，即更优的伙伴集合对应更大的效用值。",{"name":83,"@type":74,"acceptedAnswer":84},"差值版本与比值版本能否高效求解？",{"text":85,"@type":77},"文中给出结论：差值版本与比值版本都可以在同一算法框架下通过多项式时间高效求解，只需做少量修改即可。算法利用稳定匹配的 rotation-poset 表示，并使用与每个代理相关的旋转形成链这一性质。","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":22,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":22,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":20,"slug":136},19,"General","general"]