{"paper":{"title":"A bi-criteria approximation algorithm for $k$ Means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Justin Ward, Konstantin Makarychev, Maxim Sviridenko, Yury Makarychev","submitted_at":"2015-07-15T14:15:01Z","abstract_excerpt":"We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $\\beta k > k$ clusters, and must produce a clustering with cost at most $\\alpha$ times the to the cost of the optimal set of $k$ clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor. We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee $\\alpha(\\beta)$ depending on t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04227","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}