{"paper":{"title":"Optimal $e^{(\\gamma+o(1))n}$-Approximation of the Permanent of Positive Semidefinite Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Farzam Ebrahimnejad, Nima Anari","submitted_at":"2026-05-21T03:25:29Z","abstract_excerpt":"We determine, up to lower-order terms in the exponent, the best possible deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix. If $A\\succeq 0$ has no zero diagonal entry, $d=\\operatorname{rank}(A)$, $A=VV^\\dagger$ with $V\\in\\mathbb{C}^{n\\times d}$ full column rank, and $v_1,\\ldots,v_n$ are the rows of $V$, define \\[\n  \\Phi(V)=\\max_{X\\succ 0}\n  \\left\\{\\sum_{i=1}^n \\log(v_i^\\dagger Xv_i)+\\log\\det X-\\operatorname{tr} X+d\\right\\},\n  \\qquad\n  \\widehat P(A)=e^{\\Phi(V)}. \\] We prove the exact sandwich \\[\n  e^{-\\gamma n}\\widehat P(A)\\le \\opera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21946","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.21946/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}