{"type":"ScholarlyArticle","title":"The Complexity of Tournament Fixing: Subset FAS Number and Acyclic Neighborhoods","authors":[{"name":"Yuxi Liu","slug":"yuxi-liu","given":"Yuxi","family":"Liu","affiliation":""},{"name":"Junqiang Peng","slug":"junqiang-peng","given":"Junqiang","family":"Peng","affiliation":""},{"name":"Mingyu Xiao","slug":"mingyu-xiao","given":"Mingyu","family":"Xiao","affiliation":""}],"abstract":"The Tournament Fixing Problem (TFP) asks whether a knockout tournament can be scheduled to guarantee that a given player v* wins. Although TFP is NP-hard in general, it is known to be fixed-parameter tractable (FPT) when parameterized by the feedback arc/vertex set number, or the in/out-degree of v*. However, it remained open whether TFP is FPT with respect to the subset FAS number of v* --- the minimum number of arcs intersecting all cycles containing v* --- a parameter that is never larger than the aforementioned ones. In this paper, we resolve this question negatively by proving that TFP stays NP-hard even when the subset FAS number of v* is constant ≥ 1 and either the subgraph induced by the in-neighbors D[N_{in}(v*)] or the out-neighbors D[N_{out}(v*)] is acyclic. Conversely, when both D[N_{in}(v*)] and D[N_{out}(v*)] are acyclic, we show that TFP becomes FPT parameterized by the subset FAS number of v*. Furthermore, we provide sufficient conditions under which v* can win even when this parameter is unbounded.","doi":"10.24963/ijcai.2026/389","url":"https://ijcai.org/proceedings/2026/389","pdf":"https://ijcai.org/proceedings/2026/0389.pdf","markdown":"https://ijcai.org/proceedings/2026/389.md","bibtex":"https://ijcai.org/proceedings/2026/bibtex/389","volume":{"slug":"2026","title":"Proceedings of the Thirty-Fifth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2026","url":"https://ijcai.org/proceedings/2026/","year":2026,"location":"Bremen, Germany","dates":"15-21 August 2026","editors":"Diego Calvanese","isbn":"978-1-956792-09-6","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2026"},"section":"Game Theory and Economic Paradigms","track":"Main Track","number":389,"pages":{"first":3497,"last":3504},"keywords":["Game Theory and Economic Paradigms: Computational social choice"],"videos":[],"references":[],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}