{"paper":{"title":"Spectral Dual Fitting for $k$-Means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Aditya Anand, Amatya Sharma, Ernest van Wijland, Euiwoong Lee, Fabrizio Grandoni, Moses Charikar, Ruiquan Gao, Vincent Cohen-Addad","submitted_at":"2026-07-16T07:21:14Z","abstract_excerpt":"We give a new dual fitting algorithm which gives improved approximation ratios of $3+\\ln 2 + \\epsilon\\ (\\approx 3.694)$ and $4.9+\\epsilon$ for $k$-Means in (high-dimensional) Euclidean and general metrics respectively, improving upon the previously known ratios of $4+\\epsilon$ [Charikar, Cohen-Addad, Gao, Grandoni, Lee, and van Wijland STOC'26] and $5+\\epsilon$ [Byrka, Guo, Hu, Li, Wan, Wang FOCS'26], resp. In particular, our result for Euclidean $k$-Means breaks the hardness barrier of $1+8/e\\approx 3.94$ for Metric $k$-Means. Prior to our work, no such separation between general and Euclidea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14654","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.14654/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}