[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82307-en":3,"doc-seo-82307-105":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},82307,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Subexponential Algorithm for High Multiplicity Fair Division of Mixed Instances via Stereometry","Study envy-free (EF) allocation of m indivisible items among n agents when items have three types. Each agent uses additive valuations over types that can be goods (positive), chores (negative), or mixed, excluding only zero valuation vectors. Present the first subexponential-time algorithm running in (n·m)^{O(√n)} to find an EF allocation whenever it exists, otherwise correctly report impossibility. The method represents EF allocations as convex polyhedra in R^3 and applies planar cycle-separator recursion, extending to instances with fixed agents while preserving envy-freeness.","SUBEXPONENTIAL ALGORITHM FOR HIGH MULTIPLICITY FAIR DIVISION OF MIXED INSTANCES VIA STEREOMETRY  \nYuriy Dementiev  \nITMO University  \nFedor Pribytkov  \nSt. Petersburg State University  \nDanil Sagunov  \nITMO University  \narXiv :2607 .09327v 1 [ cs .DS] 10 Jul 2026  \nABSTRACT  \nWe study the problem of computing an envy-free (EF) allocation of m indivisible items among n agents when items come in three distinct types. Each agent holds additive valuations over item types that may be positive (goods), negative (chores), or mixed. We present the first subexponentialtime algorithm with running time time (n · m)O ( √n) that finds an EF allocation whenever one exists, or correctly reports that none exists. Our approach exploits a geometric representation of EF allocations as convex polyhedra in R3 and applies Miller’s planar cycle-separator theorem to recursively decompose the agent set into balanced subgroups. We further extend the algorithm to handle agents whose allocations are fixed in advance, preserving envy-freeness across all agents.  \n1 Introduction  \nFair division of resources among self-interested agents is a foundational problem in artificial intelligence and multi-agent systems [17, 8] . Among fairness criteria, envy-freeness (EF) is one of the most compelling: an allocation is envy-free if every agent weakly prefers their own bundle to that of any other agent. While EF allocations always exist for divisible goods [19], the indivisible setting is significantly more challenging—an EF allocation may not exist at all, and deciding whether one exists is computationally demanding in general.  \nMost work on envy-free allocation of indivisible goods either imposes special structural assumptions (e.g., identical valuations, binary preferences, two agents) or relaxes the fairness requirement to approximation notions such as EF1 [14, 10](envy-freeness up to one item) or EFX [11, 13] . Exact EF for general additive valuations over indivisible goods is known to be computationally hard [7], and practical algorithms typically resort to exhaustive search, which is exponential in the number of agents.  \nIn this paper we study a natural and practically motivated special case: items come in t types, and each agent has additive valuations that depend only on the type of an item rather than on individual items. This captures scenarios such as computing resource allocation across hardware tiers, food distribution by nutritional category, or budget allocation across departments. Crucially, our setting encompasses not only goods (positive valuations) but also chores (negative valuations) and mixed instances in which some item types are goods for certain agents and chores for others. The only constraint is we forbid agents with zero valuation vectors. The EF condition is uniform across all cases: every agent must weakly prefer their own bundle over anyone else’s. Despite the restriction to a fixed number of types, the problem remains non-trivial: the space of feasible allocations has size ((m1 + 1) · ... · (mt + 1))n , where mi is the total number of items of type i, rendering brute-force search infeasible for large inputs.  \nThe number of types t turns out to be the key parameter governing tractability, and the known results reveal a striking complexity landscape.  \nFor two types (t = 2), was shown that an EF allocation can be found in polynomial time. For many types (t = Ω(m)), the problem is hard even for two agents: exact EF with two agents reduces to a variant of SUBSET SUM, and an ETH-based lower bound [12] implies that no 2o (m)-time algorithm exists. In particular, the gap between t = 2 and t = Ω(m) cannot be closed without a breakthrough on ETH-hard problems.  \nThe case t = 3 studied in this paper lies between these two extremes and requires fundamentally new techniques. For the special case of a constant number of types, Maximum Nash Welfare allocations can be computed in polynomial time [18]; however, this does not yield exact EF algor","cbCaihiZTrUMvJOh","https://ap.wps.com/l/cbCaihiZTrUMvJOh","pdf",373861,1,12,"English","en",105,"# Introduction\n## Envy-freeness and computational difficulty\n## Multi-type setting and complexity landscape\n## Contribution and geometric approach\n## Related work","[{\"question\":\"What fairness criterion does the paper focus on, and when is it satisfied?\",\"answer\":\"The paper focuses on envy-freeness (EF). An allocation is EF if every agent weakly prefers its own bundle to the bundle of any other agent.\"},{\"question\":\"How are item valuations modeled in the three-type setting?\",\"answer\":\"Items come in three types, and each agent’s valuations are additive over types rather than individual items. Valuations for a type can be positive (goods), negative (chores), or mixed across agents.\"},{\"question\":\"What is the main algorithmic contribution and its running time?\",\"answer\":\"The paper presents the first subexponential-time algorithm for exact EF allocation with three types, with running time (n·m)^{O(√n)}. It either constructs an EF allocation or reports that none exists.\"}]",1784179506,30,{"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},"subexponential-algorithm-for-high-multiplicity-fair-division-of-mixed-instances-via-stereometry","",{"@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/subexponential-algorithm-for-high-multiplicity-fair-division-of-mixed-instances-via-stereometry/82307/",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 fairness criterion does the paper focus on, and when is it satisfied?","Question",{"text":75,"@type":76},"The paper focuses on envy-freeness (EF). An allocation is EF if every agent weakly prefers its own bundle to the bundle of any other agent.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are item valuations modeled in the three-type setting?",{"text":80,"@type":76},"Items come in three types, and each agent’s valuations are additive over types rather than individual items. Valuations for a type can be positive (goods), negative (chores), or mixed across agents.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main algorithmic contribution and its running time?",{"text":84,"@type":76},"The paper presents the first subexponential-time algorithm for exact EF allocation with three types, with running time (n·m)^{O(√n)}. 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