{"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 - \\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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.5765","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"}