{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:7CFSICHEJHAZD37HP63ARP4JYW","short_pith_number":"pith:7CFSICHE","canonical_record":{"source":{"id":"1407.6609","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-07-24T15:09:23Z","cross_cats_sorted":[],"title_canon_sha256":"1785567324021334df1f388912233fe63cfcadf32a496e33b79f302eea1ac9fc","abstract_canon_sha256":"e7d22aeef00b4f6b17386cedd439e29c1df38ebd5a98f859739492e6e59cd768"},"schema_version":"1.0"},"canonical_sha256":"f88b2408e449c191efe77fb608bf89c599fc1e94560a6397742cd626146848f3","source":{"kind":"arxiv","id":"1407.6609","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6609","created_at":"2026-05-18T01:20:50Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6609v3","created_at":"2026-05-18T01:20:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6609","created_at":"2026-05-18T01:20:50Z"},{"alias_kind":"pith_short_12","alias_value":"7CFSICHEJHAZ","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"7CFSICHEJHAZD37H","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"7CFSICHE","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:7CFSICHEJHAZD37HP63ARP4JYW","target":"record","payload":{"canonical_record":{"source":{"id":"1407.6609","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-07-24T15:09:23Z","cross_cats_sorted":[],"title_canon_sha256":"1785567324021334df1f388912233fe63cfcadf32a496e33b79f302eea1ac9fc","abstract_canon_sha256":"e7d22aeef00b4f6b17386cedd439e29c1df38ebd5a98f859739492e6e59cd768"},"schema_version":"1.0"},"canonical_sha256":"f88b2408e449c191efe77fb608bf89c599fc1e94560a6397742cd626146848f3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:50.702436Z","signature_b64":"LnZhSjqQ3MpUsDfNXLDxzLXn4ZsxaKsZvHLXnw0g6f5q7yXYPUaZNmCMMAuslpNg7MbdTAjMM9E79Na9O8ZGBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f88b2408e449c191efe77fb608bf89c599fc1e94560a6397742cd626146848f3","last_reissued_at":"2026-05-18T01:20:50.702071Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:50.702071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1407.6609","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:20:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zk4ZtzO/U6cYdzyf6hzcauJ9hiatgG6kPkF/PLys0GeX1ClpGmajiVRIJvViCwDN8v72L6zZVVZ1ausID/vxAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T06:57:35.217072Z"},"content_sha256":"fd69cafb628b3f9a22254762155e928e66351e329b66eef35941622c440b5bf3","schema_version":"1.0","event_id":"sha256:fd69cafb628b3f9a22254762155e928e66351e329b66eef35941622c440b5bf3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:7CFSICHEJHAZD37HP63ARP4JYW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sorting Network Relaxations for Vector Permutation Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Cong Han Lim, Stephen J. Wright","submitted_at":"2014-07-24T15:09:23Z","abstract_excerpt":"The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked in formulating relaxations of optimization problems over permutations. The Birkhoff polytope is represented using $\\Theta(n^2)$ variables and constraints, significantly more than the $n$ variables one could use to represent a permutation as a vector. Using a recent construction of Goemans (2010), we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to $\\Theta(n \\log n)$ in theory and $\\Theta(n \\log^2 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6609","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:20:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VOSKgKrpQ/HgMwdc5w0XEI1VXhyHMQLQCy7Onp7eEUZUAuqahwsNhEiqp2wolGPBllJELdKDeC4AWPrU9MghCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T06:57:35.218103Z"},"content_sha256":"94edf8ceb4bdc13427a356ef39083ca655848a02c3892809284058401bb57092","schema_version":"1.0","event_id":"sha256:94edf8ceb4bdc13427a356ef39083ca655848a02c3892809284058401bb57092"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7CFSICHEJHAZD37HP63ARP4JYW/bundle.json","state_url":"https://pith.science/pith/7CFSICHEJHAZD37HP63ARP4JYW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7CFSICHEJHAZD37HP63ARP4JYW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-04T06:57:35Z","links":{"resolver":"https://pith.science/pith/7CFSICHEJHAZD37HP63ARP4JYW","bundle":"https://pith.science/pith/7CFSICHEJHAZD37HP63ARP4JYW/bundle.json","state":"https://pith.science/pith/7CFSICHEJHAZD37HP63ARP4JYW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7CFSICHEJHAZD37HP63ARP4JYW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:7CFSICHEJHAZD37HP63ARP4JYW","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":"e7d22aeef00b4f6b17386cedd439e29c1df38ebd5a98f859739492e6e59cd768","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-07-24T15:09:23Z","title_canon_sha256":"1785567324021334df1f388912233fe63cfcadf32a496e33b79f302eea1ac9fc"},"schema_version":"1.0","source":{"id":"1407.6609","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.6609","created_at":"2026-05-18T01:20:50Z"},{"alias_kind":"arxiv_version","alias_value":"1407.6609v3","created_at":"2026-05-18T01:20:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.6609","created_at":"2026-05-18T01:20:50Z"},{"alias_kind":"pith_short_12","alias_value":"7CFSICHEJHAZ","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"7CFSICHEJHAZD37H","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"7CFSICHE","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:94edf8ceb4bdc13427a356ef39083ca655848a02c3892809284058401bb57092","target":"graph","created_at":"2026-05-18T01:20:50Z","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":"The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked in formulating relaxations of optimization problems over permutations. The Birkhoff polytope is represented using $\\Theta(n^2)$ variables and constraints, significantly more than the $n$ variables one could use to represent a permutation as a vector. Using a recent construction of Goemans (2010), we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of variables and constraints to $\\Theta(n \\log n)$ in theory and $\\Theta(n \\log^2 ","authors_text":"Cong Han Lim, Stephen J. Wright","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-07-24T15:09:23Z","title":"Sorting Network Relaxations for Vector Permutation Problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6609","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:fd69cafb628b3f9a22254762155e928e66351e329b66eef35941622c440b5bf3","target":"record","created_at":"2026-05-18T01:20:50Z","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":"e7d22aeef00b4f6b17386cedd439e29c1df38ebd5a98f859739492e6e59cd768","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2014-07-24T15:09:23Z","title_canon_sha256":"1785567324021334df1f388912233fe63cfcadf32a496e33b79f302eea1ac9fc"},"schema_version":"1.0","source":{"id":"1407.6609","kind":"arxiv","version":3}},"canonical_sha256":"f88b2408e449c191efe77fb608bf89c599fc1e94560a6397742cd626146848f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f88b2408e449c191efe77fb608bf89c599fc1e94560a6397742cd626146848f3","first_computed_at":"2026-05-18T01:20:50.702071Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:50.702071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LnZhSjqQ3MpUsDfNXLDxzLXn4ZsxaKsZvHLXnw0g6f5q7yXYPUaZNmCMMAuslpNg7MbdTAjMM9E79Na9O8ZGBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:50.702436Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.6609","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fd69cafb628b3f9a22254762155e928e66351e329b66eef35941622c440b5bf3","sha256:94edf8ceb4bdc13427a356ef39083ca655848a02c3892809284058401bb57092"],"state_sha256":"d98b2d43d6395e111d0b72f75f970967075db242e9be48f5efdbbca0817f54b2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uWJCNbo6yW1myN9FlFcXFNOF7rqSru1CG3X84KEzMskXpHYm6ePiVG3m0nkXkNyVgE6ySpVkR8j98EnpCRxTCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T06:57:35.225685Z","bundle_sha256":"324a62c2865e38dda5a36a0676648d1788674ad31aef272aec291984459b8961"}}