{"type":"ScholarlyArticle","title":"Logarithmic Approximations for Fair k-Set Selection","authors":[{"name":"Shi Li","slug":"shi-li","given":"Shi","family":"Li","affiliation":""},{"name":"Chenyang Xu","slug":"chenyang-xu","given":"Chenyang","family":"Xu","affiliation":""},{"name":"Ruilong Zhang","slug":"ruilong-zhang","given":"Ruilong","family":"Zhang","affiliation":""}],"abstract":"We study the fair k-set selection problem where we aim to select k sets from a given set system such that the (weighted) occurrence times that each element appears in these k selected sets are balanced, i.e., the maximum (weighted) occurrence times are minimized. By observing that a set system can be formulated into a bipartite graph G:=(L cup R, E), our problem is equivalent to selecting k vertices from R such that the maximum (weighted) number selected neighbors of vertices in L is minimized. The problem arises in a wide range of applications in various fields, such as machine learning, artificial intelligence, and operations research.\r\n\r\nWe first prove that the problem is NP-hard even if the maximum degree Delta of the input bipartite graph is 3, and the problem is in P when Delta=2. We then show that the problem is also in P when the input set system forms a laminar family. Based on intuitive linear programming, we show that two rounding algorithms achieve O(log n/(log log n))-approximation on general bipartite graphs, and an independent rounding algorithm achieves O(log(Delta))-approximation on bipartite graphs with a maximum degree Delta. We demonstrate that our analysis is almost tight by providing a hard instance for this linear programming.","doi":"10.24963/ijcai.2025/439","url":"https://ijcai.org/proceedings/2025/439","pdf":"https://ijcai.org/proceedings/2025/0439.pdf","markdown":"https://ijcai.org/proceedings/2025/439.md","bibtex":"https://ijcai.org/proceedings/2025/bibtex/439","volume":{"slug":"2025","title":"Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2025","url":"https://ijcai.org/proceedings/2025/","year":2025,"location":"Montreal, Canada","dates":"16-22 August 2025","editors":"James Kwok","isbn":"978-1-956792-06-5","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2025"},"section":"Game Theory and Economic Paradigms","track":"Main Track","number":439,"pages":{"first":3943,"last":3951},"keywords":["Constraint Satisfaction and Optimization: CSO: Mixed discrete and continuous optimization","Agent-based and Multi-agent Systems: MAS: Resource allocation","Game Theory and Economic Paradigms: GTEP: Computational social choice","Game Theory and Economic Paradigms: GTEP: Fair division"],"videos":[],"references":[],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}