{"paper":{"title":"Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Chaitanya Swamy, Sara Ahmadian","submitted_at":"2016-08-04T21:12:49Z","abstract_excerpt":"We consider clustering problems with {\\em non-uniform lower bounds and outliers}, and obtain the {\\em first approximation guarantees} for these problems. We have a set $\\F$ of facilities with lower bounds $\\{L_i\\}_{i\\in\\F}$ and a set $\\D$ of clients located in a common metric space $\\{c(i,j)\\}_{i,j\\in\\F\\cup\\D}$, and bounds $k$, $m$. A feasible solution is a pair $\\bigl(S\\sse\\F,\\sigma:\\D\\mapsto S\\cup\\{\\mathsf{out}\\}\\bigr)$, where $\\sigma$ specifies the client assignments, such that $|S|\\leq k$, $|\\sigma^{-1}(i)|\\geq L_i$ for all $i\\in S$, and $|\\sigma^{-1}(\\mathsf{out})|\\leq m$. In the {\\em low"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01700","kind":"arxiv","version":3},"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"}