{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ZGUONFSI2VR7UUT55MC2JRNAVL","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":"d12609583a082bf78ca951ba14eecf556e005f5ec0abbdf0a413f1f50b14d897","cross_cats_sorted":["cs.DS","math.CO","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2009-10-30T04:49:10Z","title_canon_sha256":"0ae3805e179cc5977719db9bb36379776f36de29c048e53bfcfae6e2c53c39f2"},"schema_version":"1.0","source":{"id":"0910.5765","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.5765","created_at":"2026-05-18T04:40:54Z"},{"alias_kind":"arxiv_version","alias_value":"0910.5765v3","created_at":"2026-05-18T04:40:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.5765","created_at":"2026-05-18T04:40:54Z"},{"alias_kind":"pith_short_12","alias_value":"ZGUONFSI2VR7","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"ZGUONFSI2VR7UUT5","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"ZGUONFSI","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:1681848653037485be509aefde6637b9107edfdfa2db03d0119ce3f4bcabb36b","target":"graph","created_at":"2026-05-18T04:40:54Z","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":"Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint (SDP_n) is\n   maximize \\sum_{i=1}^m \\sum_{j=1}^m A_{ij} x_i \\cdot x_j, where x_1, ..., x_m \\in S^{n-1}.\n  In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation ratio of\n  \\gamma(n) = \\frac{2}{n}(\\frac{\\Gamma((n+1)/2)}{\\Gamma(n/2)})^2 = 1 - \\Theta(1/n).\n  We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial tim","authors_text":"Fernando Mario de Oliveira Filho, Frank Vallentin, Jop Briet","cross_cats":["cs.DS","math.CO","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2009-10-30T04:49:10Z","title":"The positive semidefinite Grothendieck problem with rank constraint"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.5765","kind":"arxiv","version":3},"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":"sha256:74b5f36c48700f80c2d33759a390339445d9ded09e3735194541c996543d3be1","target":"record","created_at":"2026-05-18T04:40:54Z","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":"d12609583a082bf78ca951ba14eecf556e005f5ec0abbdf0a413f1f50b14d897","cross_cats_sorted":["cs.DS","math.CO","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2009-10-30T04:49:10Z","title_canon_sha256":"0ae3805e179cc5977719db9bb36379776f36de29c048e53bfcfae6e2c53c39f2"},"schema_version":"1.0","source":{"id":"0910.5765","kind":"arxiv","version":3}},"canonical_sha256":"c9a8e69648d563fa527deb05a4c5a0aad8763ee89705680b10909b2846595cc8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9a8e69648d563fa527deb05a4c5a0aad8763ee89705680b10909b2846595cc8","first_computed_at":"2026-05-18T04:40:54.642551Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:54.642551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S2fA0MXZJYVrSIZBHPTHh48vFZmD02W8ikCGKQLJx+CzqGFppEFeOgD93uWo9gxCx4Aeotbs+mcTn+qUhOMgAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:54.643057Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.5765","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74b5f36c48700f80c2d33759a390339445d9ded09e3735194541c996543d3be1","sha256:1681848653037485be509aefde6637b9107edfdfa2db03d0119ce3f4bcabb36b"],"state_sha256":"dcc5ace3a2095be19aec52f75609b6471df79e94e7a68f04726d6beeecd1cd51"}