{"type":"ScholarlyArticle","title":"The Computational Complexity of Almost Stable Clustering with Penalties","authors":[{"name":"Farnam Mansouri","slug":"farnam-mansouri","given":"Farnam","family":"Mansouri","affiliation":""},{"name":"Sandra Zilles","slug":"sandra-zilles","given":"Sandra","family":"Zilles","affiliation":""},{"name":"Kamyar Khodamoradi","slug":"kamyar-khodamoradi","given":"Kamyar","family":"Khodamoradi","affiliation":""}],"abstract":"We investigate the complexity of stable (or perturbation-resilient) instances of k-Means and k-Median clustering problems in metrics with small doubling dimension. While these problems have been extensively studied under multiplicative perturbation resilience in low-dimensional Euclidean spaces, we adopt a more general notion of stability, termed \"almost stable\", known from the literature as (alpha,epsilon)-perturbation resilience. Additionally, we extend our results to k-Means/k-Median with penalties, where each data point is either assigned to a cluster centre or incurs a penalty.\r\nWe show that certain special cases of almost stable k-Means/k-Median (with penalties) are solvable in polynomial time. To complement this, we also examine the hardness of almost stable instances and (1 + 1/poly(n))-stable instances of k-Means/k-Median (with penalties), proving super-polynomial lower bounds on the runtime of any exact algorithm under the widely believed Exponential Time Hypothesis (ETH).","doi":"10.24963/ijcai.2026/517","url":"https://ijcai.org/proceedings/2026/517","pdf":"https://ijcai.org/proceedings/2026/0517.pdf","markdown":"https://ijcai.org/proceedings/2026/517.md","bibtex":"https://ijcai.org/proceedings/2026/bibtex/517","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":"Machine Learning","track":"Main Track","number":517,"pages":{"first":4643,"last":4650},"keywords":["Machine Learning: Clustering","Machine Learning: Learning theory"],"videos":[],"references":[],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}