{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:KKRWS76XVYPKCK4VBSGXHXKTU7","short_pith_number":"pith:KKRWS76X","schema_version":"1.0","canonical_sha256":"52a3697fd7ae1ea12b950c8d73dd53a7d43df95c35099c287d1e91b28d13ecb3","source":{"kind":"arxiv","id":"1608.01700","version":3},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1608.01700","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-08-04T21:12:49Z","cross_cats_sorted":[],"title_canon_sha256":"5570c696cbb597f8674a6475e357a06b39145c7b3e6f61c0907cbfae684b1290","abstract_canon_sha256":"efed1a7e0fa9f32eed40f0f91fc79344d1f3af9df58b376fbdfb78cba954f9a8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:00:19.100896Z","signature_b64":"y9u1y+wV1fs7LsooVqHaBkFUsV0cCXJ17QBAZK0QwYpzfPTIxYGV/1gz0YUsRZUUzmFzYbt+21NhfUafZNAABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52a3697fd7ae1ea12b950c8d73dd53a7d43df95c35099c287d1e91b28d13ecb3","last_reissued_at":"2026-05-18T01:00:19.100173Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:00:19.100173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.01700","created_at":"2026-05-18T01:00:19.100291+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.01700v3","created_at":"2026-05-18T01:00:19.100291+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01700","created_at":"2026-05-18T01:00:19.100291+00:00"},{"alias_kind":"pith_short_12","alias_value":"KKRWS76XVYPK","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"KKRWS76XVYPKCK4V","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"KKRWS76X","created_at":"2026-05-18T12:30:25.849896+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7","json":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7.json","graph_json":"https://pith.science/api/pith-number/KKRWS76XVYPKCK4VBSGXHXKTU7/graph.json","events_json":"https://pith.science/api/pith-number/KKRWS76XVYPKCK4VBSGXHXKTU7/events.json","paper":"https://pith.science/paper/KKRWS76X"},"agent_actions":{"view_html":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7","download_json":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7.json","view_paper":"https://pith.science/paper/KKRWS76X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.01700&json=true","fetch_graph":"https://pith.science/api/pith-number/KKRWS76XVYPKCK4VBSGXHXKTU7/graph.json","fetch_events":"https://pith.science/api/pith-number/KKRWS76XVYPKCK4VBSGXHXKTU7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7/action/storage_attestation","attest_author":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7/action/author_attestation","sign_citation":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7/action/citation_signature","submit_replication":"https://pith.science/pith/KKRWS76XVYPKCK4VBSGXHXKTU7/action/replication_record"}},"created_at":"2026-05-18T01:00:19.100291+00:00","updated_at":"2026-05-18T01:00:19.100291+00:00"}