[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81670-en":3,"doc-seo-81670-105":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},81670,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Strategic Facility Location with p-Norm Social Costs","Strategic facility location is studied in ℓq(Rd) metric spaces when social cost is defined as an arbitrary p-norm of individual costs. While deterministic strategyproof mechanisms have strong approximation results in one-dimensional settings, multi-dimensional guarantees under general p-norm objectives remain less developed. This work analyzes the coordinate-wise median (CM) mechanism and proves approximation bounds for generalized social cost norms. It shows CM remains within ratio 3 for all monotone symmetric norms, and derives tight bounds for d=2 and refined dimension-dependent relationships for d≥3.","arXiv :2606 . 12 187v2 [ cs .GT] 10 Jul 2026  \nStrategic Facility Location with p-Norm Social Costs  \nJabari Hastings*  \nStanford University  \nAbstract  \nWe consider the strategic facility location problem in ℓq (Rd ) spaces where the social cost is defined by an arbitrary p-norm of the individual costs. While the optimal approximation ratios for deterministic strategyproof mechanisms are well established in the d = 1 setting, the guarantees for multi-dimensional spaces under an arbitrary p-norm are less understood.  \nIn this work, we analyze the well-studied, strategyproof coordinate-wise median (CM) mechanism and provide approximation guarantees for these generalized social costs.  \n• We show that the CM mechanism is in fact robust to a broader class of social objectives: for every monotone symmetric norm objective, including all p-norm social costs, its approximation ratio never exceeds 3 in arbitrary ℓq (Rd ) spaces, regardless of the dimension.  \n• For d = 2, we establish tight approximation ratios for all p, q ≥ 1 in ℓq (R2 ) . In particular, we show that the CM mechanism is a 21−1/ max(p,q)-approximation, resolving the conjecture of Goel and Hann-Caruthers (Social Choice and Welfare, 2023) in the Euclidean case and extending the guarantee to arbitrary ℓq distances.  \n• For d ≥ 3, we refine the dimension-independent approximation guarantee for p-norm social costs in ℓq (Rd ) spaces, giving upper bounds that depend on the relationship between the social-cost norm p and the underlying distance norm q. This generalizes the recent result of Gravin and Jia (STOC, 2025) for the utilitarian social cost.  \n*Supported by the Simons Foundation Collaboration on the Theory of Algorithmic Fairness and the Simons Foundation Investigators Award 17351 .  \nContents  \n1 Introduction 3  \n1.1 Our Contributions ...................................... 4  \n1.2 Related Work ......................................... 6  \n2 Preliminaries 7  \n2.1 Notation and Terminology .................................. 7  \n2.2 Coordinate-wise Median ................................... 8  \n3 Monotone Symmetric Norm Objectives in Rd 9  \n4 A Program for p-Norm Social Costs 12  \n5 p-Norm Social Costs in ℓq (R2 ) 12  \n5.1 Optimization over Opposing Orthants ........................... 13  \n5.2 Approximation Ratio ..................................... 14  \n6 p-Norm Social Costs in ℓq (Rd ) 16  \n6.1 Upper Bounds on Approximation Ratio .......................... 17  \n6.1.1 Optimization over Individual Points ........................ 17  \n6.1.2 Program Relaxation ................................. 19  \n6.1.3 The Case of p/q ∈ {1, 2} ............................... 20  \n6.1.4 Characterizing the Remaining Cases ........................ 21  \n6.1.5 The Case of p/q ∈ (1, 2) ............................... 21  \n6.1.6 The Case of p/q ∈ (2, ∞) .............................. 25  \n6.1.7 The Case of p/q ∈ (0, 1) ............................... 28  \n6.1.8 Proof of Theorem 3 .................................. 31  \n7 Conclusion 32  \n7.1 Discussion and Future Directions .............................. 32  \n7.2 Acknowledgements ...................................... 32  \nA Proofs from Section 6 36  \nA.1 Proof of Lemma 8 ....................................... 36  \nA.2 Proof of Lemma 10 ...................................... 41  \nA.3 Proof of Lemma 11 ...................................... 42  \nA.4 Proof of Lemma 13 and Lemma 15 ............................. 43  \n1 Introduction  \nFacility location is a fundamental problem at the intersection of combinatorial optimization and mechanism design. In its canonical form, there are n agents positioned at locations x1, . . . , xn within a metric space, and the goal is to select a facility location f that (approximately) minimizesa social cost derived from the distances between the agents and the facility. Within the optimization community, research has focused extensively on generalizations of the problem, including multiple facilities (Hochbaum an","cbCaibZPpBZPmovG","https://ap.wps.com/l/cbCaibZPpBZPmovG","pdf",327373,3,1,44,"English","en",105,"# Introduction\n## Our Contributions\n## Related Work\n# Preliminaries\n## Notation and Terminology\n## Coordinate-wise Median\n# Monotone Symmetric Norm Objectives in Rd\n# A Program for p-Norm Social Costs\n# p-Norm Social Costs in ℓq(R2)\n## Optimization over Opposing Orthants\n## Approximation Ratio\n# p-Norm Social Costs in ℓq(Rd)\n## Upper Bounds on Approximation Ratio\n## Optimization over Individual Points\n## Program Relaxation\n## The Case of p/q ∈ {1, 2}\n## Characterizing the Remaining Cases\n## The Case of p/q ∈ (1, 2)\n## The Case of p/q ∈ (2, ∞)\n## The Case of p/q ∈ (0, 1) \n# Conclusion\n## Discussion and Future Directions\n## Acknowledgements","[{\"question\":\"What problem does the paper study in strategic facility location?\",\"answer\":\"It studies facility location where agents may strategically misreport locations, and the goal is to choose a facility that minimizes (approximately) a social cost defined by an arbitrary p-norm of individual costs in ℓq(Rd) spaces.\"},{\"question\":\"Which mechanism is analyzed and what does it guarantee?\",\"answer\":\"The paper analyzes the coordinate-wise median (CM) mechanism, proving approximation guarantees for generalized social cost norms under strategyproofness constraints.\"},{\"question\":\"What approximation bounds are obtained for the CM mechanism in different dimensions?\",\"answer\":\"For any dimension and any monotone symmetric norm objective (including all p-norm social costs), CM’s approximation ratio never exceeds 3. For d=2, tight approximation ratios are established for all p,q≥1, and for d≥3 the bounds are refined based on the relationship between p and q.\"}]",1784175319,111,{"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},"strategic-facility-location-with-p-norm-social-costs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/strategic-facility-location-with-p-norm-social-costs/81670/",4,{"url":51,"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","2026-07-24","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 problem does the paper study in strategic facility location?","Question",{"text":75,"@type":76},"It studies facility location where agents may strategically misreport locations, and the goal is to choose a facility that minimizes (approximately) a social cost defined by an arbitrary p-norm of individual costs in ℓq(Rd) spaces.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which mechanism is analyzed and what does it guarantee?",{"text":80,"@type":76},"The paper analyzes the coordinate-wise median (CM) mechanism, proving approximation guarantees for generalized social cost norms under strategyproofness constraints.",{"name":82,"@type":73,"acceptedAnswer":83},"What approximation bounds are obtained for the CM mechanism in different dimensions?",{"text":84,"@type":76},"For any dimension and any monotone symmetric norm objective (including all p-norm social costs), CM’s approximation ratio never exceeds 3. For d=2, tight approximation ratios are established for all p,q≥1, and for d≥3 the bounds are refined based on the relationship between p and q.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]