[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82577-en":3,"doc-seo-82577-105":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},82577,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Fair Allocation under Conflict Constraints via Strong Colorability","Fair allocation under conflict constraints partitions the vertices of a graph among agents so that adjacent vertices are never assigned to the same agent, while maintaining fairness. The work studies common-preference agents under three envy-freeness notions: SD-EF1 (stochastic dominance up to one item), EF1 (additive valuations up to one item), and EF[1,1] (up to one item from each side). It links the problem to the hierarchy of the strong chromatic number, yielding characterization and sufficient-condition results, with tight guarantees driven by maximum degree.","arXiv :2607 .0 1059v 1 [ cs .GT] 1 Jul 2026  \nFair Allocation under Conflict Constraints via Strong Colorability  \nIshay Haviv*  \nAbstract  \nIn the fair allocation problem under conflict constraints, the goal is to partition the vertices of a graph among agents in a fair manner, such that no two adjacent vertices are assigned to the same agent. We study this problem for agents with common preferences through the lens of three fairness criteria: stochastic-dominance envy-freeness up to one item for preference orders (SD-EF1), envy-freeness up to one item for monotone additive valuations (EF1), and envy-freeness up to one item from each side for general additive valuations (EF[1,1]) . To do so, we introduce a hierarchy of variants of the strong chromatic number, a graph quantity introduced independently by Alon and Fellows in the early nineties. Our results reveal a close connection between fair allocation under conflict constraints and the first two levels of this hierarchy, providing a unified route to both existential and algorithmic results.  \nFor SD-EF1, we fully characterize the number of agents needed to guarantee a fair allocation of a given graph for every common preference order. For EF1 and EF[1,1], we provide analogous sufficient conditions, extending a result on path graphs due to Equbal, Gurjar, Igarashi, Kumar, Manurangsi, Nath, Saxena, Vaish, andYoneda. We also show that, unlike in the SD-EF1 setting, the sufficient conditions for EF1 and EF[1,1] are not necessary in general. Our framework yields existential and algorithmic consequences in terms of the maximum degree. We obtain that every graph with maximum degree ∆ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least 3∆ − 1. We further provide, for any fixed ε > 0, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least (3 + ε)∆ . These guarantees strengthen earlier work by Barman and Viswanathan on equitable colorings.  \n*The Academic College of Tel Aviv-Yaffo, Tel Aviv, Israel.  \n1 Introduction  \nThe fair allocation problem concerns the task of partitioning a set of indivisible resources among several agents with diverse preferences. This problem has received sustained attention across a variety of research fields, including mathematics, economics, and computer science, and arises naturally in numerous real-world applications, such as allocating computing resources among users and assigning course seats to students. In the standard setting, we are given a set M of resources, referred to as items, and a set of ℓ agents, each endowed with preferences over the subsets of M. An allocation of M to the agents is a sequence (A1, . . . , A ℓ) of pairwise disjoint sets whose union is M, where Ai denotes the set of items assigned to agent i. Given the agents’preferences, the objective is to find an allocation that satisfies a prescribed fairness criterion.  \nOne natural concept of fairness, introduced by Bogomolnaia and Moulin [10] and further developed by Aziz, Gaspers, Mackenzie, and Walsh [5], relies on the notion of stochastic dominance to compare assignments. In this setting, each agent i ∈ [ℓ] is associated with a weak order ⪰ion M, representing an ordinal ranking of the items. An allocation is called stochastic-dominance envy-free (SD-EF) if the assignment of each agent i ∈ [ℓ] stochastically dominates the assignment of any other agent under her preference order ⪰i, meaning that for every item α ∈ M, the agent’s assignment includes at least as many items weakly preferred to α under ⪰i as any other agent’s assignment does. However, this condition is often too demanding and is rarely achievable. This gives rise to the relaxation of this notion, called stochastic-dominance envy-freeness up to one item (SDEF1), which allows a hypothetical removal of at most one item from the other agent’s assignment before this comparison (see, e.g.,[6, 14]) .  \nAnother commo","cbCaifaGCCOmhDz9","https://ap.wps.com/l/cbCaifaGCCOmhDz9","pdf",337583,1,32,"English","en",105,"# Abstract\n# Introduction\n## Fair allocation and fairness notions\n## Conflict constraints and feasibility settings","[{\"question\":\"What problem does the paper study?\",\"answer\":\"The paper studies fair allocation under conflict constraints by partitioning graph vertices among agents so that no edge connects vertices assigned to the same agent, while satisfying specific fairness criteria.\"},{\"question\":\"What are the three fairness notions analyzed?\",\"answer\":\"It analyzes SD-EF1 for preference orders using stochastic dominance, EF1 for monotone additive valuations using additive comparisons up to one removed item, and EF[1,1] for general additive valuations using up to one item removed from each side.\"},{\"question\":\"How do the results depend on the graph’s maximum degree?\",\"answer\":\"The paper shows that every graph with maximum degree Δ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least 3Δ−1, and provides deterministic polynomial-time algorithms for agent counts at least (3+ε)Δ.\"}]",1784181617,81,{"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},"fair-allocation-under-conflict-constraints-via-strong-colorability","",{"@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/fair-allocation-under-conflict-constraints-via-strong-colorability/82577/",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 problem does the paper study?","Question",{"text":75,"@type":76},"The paper studies fair allocation under conflict constraints by partitioning graph vertices among agents so that no edge connects vertices assigned to the same agent, while satisfying specific fairness criteria.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are the three fairness notions analyzed?",{"text":80,"@type":76},"It analyzes SD-EF1 for preference orders using stochastic dominance, EF1 for monotone additive valuations using additive comparisons up to one removed item, and EF[1,1] for general additive valuations using up to one item removed from each side.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the results depend on the graph’s maximum degree?",{"text":84,"@type":76},"The paper shows that every graph with maximum degree Δ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least 3Δ−1, and provides deterministic polynomial-time algorithms 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