{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:NEI45SQTFNKSTN3GKL2QAR4LGT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"48a0a835f50d18f4558ed1eedf41a2779f5d5221aad01c3f2dd696e7042558ab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-02-23T17:48:35Z","title_canon_sha256":"7c9d0af76a5c584a99f1ec498f2b3ced909e70e3a060f911954428c35b110b51"},"schema_version":"1.0","source":{"id":"1502.06528","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.06528","created_at":"2026-05-18T02:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1502.06528v1","created_at":"2026-05-18T02:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.06528","created_at":"2026-05-18T02:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"NEI45SQTFNKS","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NEI45SQTFNKSTN3G","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NEI45SQT","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:0814fbe9344750a9b828520fe6c77f68776e30e2750627a7c38aae1497303f68","target":"graph","created_at":"2026-05-18T02:26:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper defines weak-$\\alpha$-supermodularity for set functions. Many optimization objectives in machine learning and data mining seek to minimize such functions under cardinality constrains. We prove that such problems benefit from a greedy extension phase. Explicitly, let $S^*$ be the optimal set of cardinality $k$ that minimizes $f$ and let $S_0$ be an initial solution such that $f(S_0)/f(S^*) \\le \\rho$. Then, a greedy extension $S \\supset S_0$ of size $|S| \\le |S_0| + \\lceil \\alpha k \\ln(\\rho/\\varepsilon) \\rceil$ yields $f(S)/f(S^*) \\le 1+\\varepsilon$. As example usages of this framewor","authors_text":"Christos Boutsidis, Edo Liberty, Maxim Sviridenko","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-02-23T17:48:35Z","title":"Greedy Minimization of Weakly Supermodular Set Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.06528","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:19d4b525bb4b1cc70bd918e9aacb82582ab38a8dfae80b08baf0ae7743735d02","target":"record","created_at":"2026-05-18T02:26:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"48a0a835f50d18f4558ed1eedf41a2779f5d5221aad01c3f2dd696e7042558ab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-02-23T17:48:35Z","title_canon_sha256":"7c9d0af76a5c584a99f1ec498f2b3ced909e70e3a060f911954428c35b110b51"},"schema_version":"1.0","source":{"id":"1502.06528","kind":"arxiv","version":1}},"canonical_sha256":"6911ceca132b5529b76652f500478b34f8e67560415cc968818e6ce0fde7ae7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6911ceca132b5529b76652f500478b34f8e67560415cc968818e6ce0fde7ae7f","first_computed_at":"2026-05-18T02:26:32.330300Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:32.330300Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s+V1Alk91Qn9RET7dTbQqWO7CYuSsOocZQH3wgwYF8gVxM5XOw/7m4wutAfHGcuIUPxAUX58BtI7kSWXno0+Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:32.330747Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.06528","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19d4b525bb4b1cc70bd918e9aacb82582ab38a8dfae80b08baf0ae7743735d02","sha256:0814fbe9344750a9b828520fe6c77f68776e30e2750627a7c38aae1497303f68"],"state_sha256":"5f5b28e3d6edbb646c92ae449b23d6e9c377b68d86b12e08764b5d38d0be5ba8"}