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We call the convex hull of the second-order permutation matrices, the {\\em second-order Birkhoff polytope} and denote it by ${\\cal B}^{[2]}$. It can be seen that ${\\cal B}^{[2]}$ is isomorphic to the QAP-polytope, the domain of optimization in {\\em quadratic assignment problem}"},"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":"1409.0667","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-09-02T11:18:53Z","cross_cats_sorted":[],"title_canon_sha256":"3147c12d6039c3680620d69dc772b42c3dcdf4dcc28850818e0ea0bd743018b3","abstract_canon_sha256":"c401756821ab2fa4de528df33f1bcb4b882225b49ea59dc508e62de331d39fc0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:25.739957Z","signature_b64":"0sabg3a/k7/yt8+GXblLoBzEDUw/GNwQpS1k050t8uTcnBDdU5e1jcGPfo2Y8YEbfZ/HWwH18qK1avgTx+HACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36756aa9b0e564dc83aa1f6b5cc3f146453f81f6acecb00550a5e5d0bce4b003","last_reissued_at":"2026-05-18T02:43:25.739402Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:25.739402Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New Facets of the QAP-Polytope","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Pawan Kumar Aurora, Shashank K Mehta","submitted_at":"2014-09-02T11:18:53Z","abstract_excerpt":"The Birkhoff polytope is defined to be the convex hull of permutation matrices, $P_{\\sigma}\\ \\forall \\sigma\\in S_n$. We define a second-order permutation matrix $P^{[2]}_{\\sigma}$ in $\\mathbb{R}^{n^2\\times n^2}$ corresponding to a permutation $\\sigma$ as $(P^{[2]}_{\\sigma})_{ij,kl} = (P_{\\sigma})_{ij}(P_{\\sigma})_{kl}$. We call the convex hull of the second-order permutation matrices, the {\\em second-order Birkhoff polytope} and denote it by ${\\cal B}^{[2]}$. It can be seen that ${\\cal B}^{[2]}$ is isomorphic to the QAP-polytope, the domain of optimization in {\\em quadratic assignment problem}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.0667","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1409.0667","created_at":"2026-05-18T02:43:25.739478+00:00"},{"alias_kind":"arxiv_version","alias_value":"1409.0667v2","created_at":"2026-05-18T02:43:25.739478+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.0667","created_at":"2026-05-18T02:43:25.739478+00:00"},{"alias_kind":"pith_short_12","alias_value":"GZ2WVKNQ4VSN","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"GZ2WVKNQ4VSNZA5K","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"GZ2WVKNQ","created_at":"2026-05-18T12:28:30.664211+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ","json":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ.json","graph_json":"https://pith.science/api/pith-number/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/graph.json","events_json":"https://pith.science/api/pith-number/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/events.json","paper":"https://pith.science/paper/GZ2WVKNQ"},"agent_actions":{"view_html":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ","download_json":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ.json","view_paper":"https://pith.science/paper/GZ2WVKNQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1409.0667&json=true","fetch_graph":"https://pith.science/api/pith-number/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/graph.json","fetch_events":"https://pith.science/api/pith-number/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/action/storage_attestation","attest_author":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/action/author_attestation","sign_citation":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/action/citation_signature","submit_replication":"https://pith.science/pith/GZ2WVKNQ4VSNZA5KD5VVZQ7RIZ/action/replication_record"}},"created_at":"2026-05-18T02:43:25.739478+00:00","updated_at":"2026-05-18T02:43:25.739478+00:00"}