{"type":"ScholarlyArticle","title":"Almost Envy-Freeness for Groups: Improved Bounds via Discrepancy Theory","authors":[{"name":"Pasin Manurangsi","slug":"pasin-manurangsi","given":"Pasin","family":"Manurangsi","affiliation":"Google Research , USA"},{"name":"Warut Suksompong","slug":"warut-suksompong","given":"Warut","family":"Suksompong","affiliation":"School of Computing , National University of Singapore , Singapore"}],"abstract":"We study the allocation of indivisible goods among groups of agents using well-known fairness notions such as envy-freeness and proportionality. While these notions cannot always be satisfied, we provide several bounds on the optimal relaxations that can be guaranteed. For instance, our bounds imply that when the number of groups is constant and the $n$ agents are divided into groups arbitrarily, there exists an allocation that is envy-free up to $\\Theta(\\sqrt{n})$ goods, and this bound is tight. Moreover, we show that while such an allocation can be found efficiently, it is NP-hard to compute an allocation that is envy-free up to $o(\\sqrt{n})$ goods even when a fully envy-free allocation exists. Our proofs make extensive use of tools from discrepancy theory.","doi":"10.24963/ijcai.2021/47","url":"https://ijcai.org/proceedings/2021/47","pdf":"https://ijcai.org/proceedings/2021/0047.pdf","markdown":"https://ijcai.org/proceedings/2021/47.md","bibtex":"https://ijcai.org/proceedings/2021/bibtex/47","volume":{"slug":"2021","title":"Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2021","url":"https://ijcai.org/proceedings/2021/","year":2021,"location":"Montreal","dates":"19-27 August 2021","editors":"Prof. Zhi-Hua Zhou","isbn":"978-0-9992411-9-6","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2021"},"section":"Agent-based and Multi-agent Systems","track":"Main Track","number":47,"pages":{"first":335,"last":341},"keywords":["Agent-based and Multi-agent Systems: Resource Allocation","Agent-based and Multi-agent Systems: Computational Social Choice"],"videos":[],"references":[{"position":1,"raw":"[Aleksandrov and Walsh, 2018] Martin Aleksandrov and Toby Walsh. 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