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\\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"},"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":"0910.5765","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2009-10-30T04:49:10Z","cross_cats_sorted":["cs.DS","math.CO","math.FA"],"title_canon_sha256":"0ae3805e179cc5977719db9bb36379776f36de29c048e53bfcfae6e2c53c39f2","abstract_canon_sha256":"d12609583a082bf78ca951ba14eecf556e005f5ec0abbdf0a413f1f50b14d897"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:54.643057Z","signature_b64":"S2fA0MXZJYVrSIZBHPTHh48vFZmD02W8ikCGKQLJx+CzqGFppEFeOgD93uWo9gxCx4Aeotbs+mcTn+qUhOMgAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9a8e69648d563fa527deb05a4c5a0aad8763ee89705680b10909b2846595cc8","last_reissued_at":"2026-05-18T04:40:54.642551Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:54.642551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The positive semidefinite Grothendieck problem with rank constraint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.CO","math.FA"],"primary_cat":"math.OC","authors_text":"Fernando Mario de Oliveira Filho, Frank Vallentin, Jop Briet","submitted_at":"2009-10-30T04:49:10Z","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 - 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