[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82158-en":3,"doc-seo-82158-105":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},82158,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Optimal Metric Distortion for Learning-Augmented Matching on the Line","Revisits matching on a line with ordinal preferences, where n agents and n items lie on an unknown shared line metric and the goal is low-cost perfect matching using only agents’ distance rankings. Distortion α is defined as a worst-case multiplicative bound on the social cost versus optimum over all consistent line metrics. In learning-augmented settings, the mechanism uses an instance prediction of uncertain quality, aiming for 1-consistency when accurate and 3-robustness under arbitrary inaccuracy.","Optimal Metric Distortion for Learning-Augmented  \nMatching on the Line  \nJabari Hastings* Stanford University  \nMarena Richter† University of Bonn  \narXiv :2607 .09038v 1 [ cs .GT] 10 Jul 2026  \nJuly 13, 2026  \nAbstract  \nWe revisit the problem of matching on the line with ordinal preferences. In the classic setting, there are n agents and n items in a shared unknown line metric, and the goal is to find a low-cost perfect matching using only the agents’ rankings of the items by distance. A mechanism has distortion α if it always outputs a matching whose cost is within a factor of α of the optimum, in every consistent line metric.  \nIn the learning-augmented setting, the mechanism is also supplied with a prediction that conveys additional information about the instance. The quality of this prediction is unknown, and the goal is to optimize the mechanism’s distortion when the prediction is accurate (consistency), while preserving worst-case guarantees when the prediction is arbitrarily inaccurate (robustness) . We propose a mechanism that takes a matching as its prediction and guarantees 1-consistency and 3-robustness. By recovering an optimal matching when the prediction is perfectly accurate while retaining the optimal prediction-free distortion guarantee when it is arbitrarily inaccurate, we resolve an open question of Filos-Ratsikas et al. (IJCAI, 2025) .  \n*Supported by the Simons Foundation Collaboration on the Theory of Algorithmic Fairness and the Simons Foundation Investigators Award 17351 .  \n†Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)– 459420781 (FOR AlgoForGe) and affiliated with the Lamarr Institute for Machine Learning and Artificial Intelligence.  \n1 Introduction  \nMatching is a fundamental assignment task that underpins a host of high-stakes, real-world scenarios. To see its profound societal impact, one needs to look no further than the school choice problem. In many countries, the placement of children into schools relies on centralized matching mechanisms. Given the outsize role of schooling in a child’s development, these assignments are frequently contentious. Describing the pressure these systems place on families, Tobin (2003) wrote of one U.S. school district:  \n“This complicated new system requires you to be a savvy parent. Some parents have treated the choice application with a casualness befitting a form for summer camp. Yet it requires the same level of care you would give an IRS tax return.”  \n—Thomas Tobin, St. Petersburg Times, 2003  \nPart of this anxiety stems from the informational constraints of the matching mechanisms often deployed, such as the Boston Mechanism (Abdulkadiro˘glu and Snmez, 2003), Deferred Acceptance (Gale and Shapley, 1962), and Random Serial Dictatorship (Abdulkadiro˘gluandSnmez, 1998) . These algorithms typically receive only ordinal information (e.g., rankings), while the cardinal social costs they are meant to optimize are often latent or under-specified. Given this limitation, it seems reasonable to expect them to be less efficient than an algorithm with complete information.  \nIn computational social choice, an ordinal mechanism’s loss in efficiency is often analyzed under the distortion framework, introduced by Procaccia and Rosenschein (2006) and surveyed by Anshelevich et al. (2021) . Here, an ordinal mechanism is evaluated in terms of its worst-case approximation ratio of the optimal social cost (called the distortion), over all cardinal values consistent with a given set of rankings. Since the approximation ratio could be unbounded when the underlying cardinal values are unrestricted, the literature often adds mild constraints to make the analysis tractable and meaningful. A particularly fruitful direction is the metric distortion model (Anshelevich et al., 2015), where ordinal preferences are assumed to arise from distances in an unknown metric space. The goal is then to design mechanisms that, using only these rankings, achieve","cbCaioUQ5Yy51SP0","https://ap.wps.com/l/cbCaioUQ5Yy51SP0","pdf",268327,1,25,"English","en",105,"# Abstract\n# Introduction\n## Matching and informational constraints\n## Distortion framework and metric distortion\n## Prior results and open challenges\n## Line metric as a starting point","[{\"question\":\"What does distortion α mean for mechanisms in matching on the line?\",\"answer\":\"A mechanism has distortion α if it always outputs a matching whose cost is within a factor of α of the optimum for every consistent line metric.\"},{\"question\":\"How does the learning-augmented setting differ from the classic ordinal-only setting?\",\"answer\":\"The mechanism additionally receives a prediction that may be accurate or inaccurate, and it must optimize distortion when predictions are consistent while keeping worst-case guarantees when predictions are wrong.\"},{\"question\":\"What performance guarantees does the proposed mechanism provide?\",\"answer\":\"The mechanism guarantees 1-consistency and 3-robustness, recovering an optimal matching under perfect prediction while retaining the optimal prediction-free distortion guarantee when predictions are arbitrarily inaccurate.\"}]",1784178505,63,{"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},"optimal-metric-distortion-for-learning-augmented-matching-on-the-line","",{"@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/optimal-metric-distortion-for-learning-augmented-matching-on-the-line/82158/",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 distortion α mean for mechanisms in matching on the line?","Question",{"text":75,"@type":76},"A mechanism has distortion α if it always outputs a matching whose cost is within a factor of α of the optimum for every consistent line metric.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the learning-augmented setting differ from the classic ordinal-only setting?",{"text":80,"@type":76},"The mechanism additionally receives a prediction that may be accurate or inaccurate, and it must optimize distortion when predictions are consistent while keeping worst-case guarantees when predictions are wrong.",{"name":82,"@type":73,"acceptedAnswer":83},"What performance guarantees does the proposed mechanism provide?",{"text":84,"@type":76},"The mechanism guarantees 1-consistency and 3-robustness, recovering an optimal matching under perfect prediction while retaining the optimal prediction-free distortion guarantee when predictions are arbitrarily 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